7112:
6499:
7107:{\displaystyle {\begin{aligned}I&={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}A&B\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\&={\begin{bmatrix}A&B\end{bmatrix}}\left({\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}{\begin{bmatrix}A&B\end{bmatrix}}\right)^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\&={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}A^{\mathsf {T}}A&O\\O&B^{\mathsf {T}}B\end{bmatrix}}^{-1}{\begin{bmatrix}A^{\mathsf {T}}\\B^{\mathsf {T}}\end{bmatrix}}\\&=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}+B\left(B^{\mathsf {T}}B\right)^{-1}B^{\mathsf {T}}\end{aligned}}}
14003:
4876:
42:
14267:
4668:
1741:
10769:
4871:{\displaystyle P_{\mathbf {u} }\mathbf {x} =\mathbf {u} \mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\parallel }+\mathbf {u} \mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\perp }=\mathbf {u} \left(\operatorname {sgn} \left(\mathbf {u} ^{\mathsf {T}}\mathbf {x} _{\parallel }\right)\left\|\mathbf {x} _{\parallel }\right\|\right)+\mathbf {u} \cdot \mathbf {0} =\mathbf {x} _{\parallel }}
4356:
4086:
7378:
1621:
11306:
1358:
10546:
7417:
A projection is defined by its kernel and the basis vectors used to characterize its range (which is a complement of the kernel). When these basis vectors are orthogonal to the kernel, then the projection is an orthogonal projection. When these basis vectors are not orthogonal to the kernel, the
10135:
3476:
4229:
3740:
3967:
3962:
3381:
7210:
9972:
1476:
1186:
8988:
5972:
11164:
10286:
6199:
457:
10446:
5153:
3112:
If two orthogonal projections commute then their product is an orthogonal projection. If the product of two orthogonal projections is an orthogonal projection, then the two orthogonal projections commute (more generally: two self-adjoint
10003:
2935:
1126:
1028:
3386:
5897:
5277:
8245:
2510:
8883:
10623:
3641:
12389:
4663:
4407:
11374:
6288:
10379:
5665:
8807:
8630:
4537:
9851:
1465:
12156:
7939:
6504:
2306:
9876:
9207:
7185:
8158:
8035:
3863:
3282:
8529:
9792:
4351:{\displaystyle \left\|P\mathbf {v} \right\|^{2}=\langle P\mathbf {v} ,P\mathbf {v} \rangle =\langle P\mathbf {v} ,\mathbf {v} \rangle \leq \left\|P\mathbf {v} \right\|\cdot \left\|\mathbf {v} \right\|}
9085:
7797:
7559:
5532:
5083:
4937:
6077:
5799:
2409:
10658:
6412:
2555:
5835:
5759:
4081:{\displaystyle \langle \mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P^{*}\mathbf {y} \rangle }
13185:
9341:
8081:
3636:
8409:
7373:{\displaystyle I={\begin{bmatrix}A&B\end{bmatrix}}{\begin{bmatrix}\left(A^{\mathsf {T}}WA\right)^{-1}A^{\mathsf {T}}\\\left(B^{\mathsf {T}}WB\right)^{-1}B^{\mathsf {T}}\end{bmatrix}}W.}
493:
4446:
3772:
7386:
is used instead of the transpose. Further details on sums of projectors can be found in
Banerjee and Roy (2014). Also see Banerjee (2004) for application of sums of projectors in basic
2442:
9434:
641:
8888:
1616:{\displaystyle P^{2}={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}{\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}=P.}
13379:
5198:
10959:
10689:
5902:
2843:
1039:
12840:
2028:
1731:
944:
6450:
3597:
1395:
13056:
11517:
11301:{\displaystyle P={\begin{bmatrix}1&\sigma _{1}\\0&0\end{bmatrix}}\oplus \cdots \oplus {\begin{bmatrix}1&\sigma _{k}\\0&0\end{bmatrix}}\oplus I_{m}\oplus 0_{s}.}
9242:
9016:
2371:
1965:
8748:
8480:
8346:
8274:
5459:
1911:
1353:{\displaystyle P^{2}{\begin{bmatrix}x\\y\\z\end{bmatrix}}=P{\begin{bmatrix}x\\y\\0\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}=P{\begin{bmatrix}x\\y\\z\end{bmatrix}}.}
897:
727:
13334:
11977:
5308:
10194:
9672:
7456:
5203:
3067:
257:
1990:
694:
10739:
10711:
10572:
10421:
8571:
8451:
7963:
6337:
6109:
4611:
4585:
4559:
4476:
4220:
3546:
3524:
3257:
3235:
1933:
1882:
398:
11733:
11598:
11436:
2343:
137:
12080:
8716:
7605:
5033:
1693:
13109:
12998:
12766:
11774:
11477:
9490:
5088:
13089:
12036:
11920:
11847:
11140:
11090:
10884:
10819:
7863:
7651:
7489:
5562:
4987:
4179:
2838:
1832:
1181:
935:
868:
566:
290:
11409:
11033:
9703:
9620:
9272:
6364:
6104:
5403:
2614:
778:
12585:
13560:
12003:
11571:
11120:
10181:
9998:
9569:
9523:
9134:
7723:
7677:
3838:
8369:
8317:
6204:
13205:
13129:
12975:
12955:
12923:
12903:
12883:
12863:
12712:
12643:
12605:
12526:
12506:
12486:
12452:
12432:
12412:
12287:
12267:
12243:
12223:
12203:
12183:
12100:
11887:
11867:
11814:
11794:
11701:
11681:
11654:
11630:
11537:
11057:
11006:
10982:
10904:
10839:
10759:
10441:
10399:
10314:
10155:
9871:
9640:
9593:
9543:
9454:
9365:
9108:
8690:
8670:
8650:
8549:
8429:
8294:
8101:
7983:
7837:
7817:
7747:
7697:
7625:
7579:
7509:
7205:
6494:
6470:
6019:
5992:
5707:
5687:
5582:
5479:
5427:
5376:
5356:
5336:
5007:
4961:
4146:
4126:
4106:
3858:
3812:
3792:
3566:
3502:
3277:
3209:
3189:
3153:
3087:
3007:
2987:
2959:
2794:
2774:
2754:
2734:
2714:
2694:
2674:
2654:
2634:
2582:
2253:
2233:
2210:
2187:
2167:
2139:
2119:
2096:
2076:
2056:
1795:
1664:
1641:
1148:
802:
751:
589:
529:
389:
369:
341:
313:
225:
177:
157:
97:
5587:
4488:
5840:
1408:
7868:
2258:
10541:{\displaystyle \operatorname {proj} _{V}\mathbf {y} ={\frac {\mathbf {y} \cdot \mathbf {u} ^{i}}{\mathbf {u} ^{i}\cdot \mathbf {u} ^{i}}}\mathbf {u} ^{i}}
8163:
2447:
7402:
is sometimes used to refer to non-orthogonal projections. These projections are also used to represent spatial figures in two-dimensional drawings (see
2969:
are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a subspace
10130:{\displaystyle \sigma _{i}={\begin{cases}{\sqrt {1+\gamma _{i}^{2}}}&1\leq i\leq k\\1&k+1\leq i\leq n-k\\0&{\text{otherwise}}\end{cases}}}
8812:
10577:
7943:
This expression generalizes the formula for orthogonal projections given above. A standard proof of this expression is the following. For any vector
12296:
12245:
must be a closed subspace. Furthermore, the kernel of a continuous projection (in fact, a continuous linear operator in general) is closed. Thus a
4616:
4361:
3471:{\displaystyle \langle \mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,P\mathbf {y} \rangle =\langle P\mathbf {x} ,\mathbf {y} \rangle .}
11314:
10319:
8753:
8576:
9797:
13265:
As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutative generalizations of
12105:
9139:
7119:
3735:{\displaystyle \langle P\mathbf {x} ,\mathbf {y} -P\mathbf {y} \rangle =\langle \mathbf {x} ,\left(P-P^{2}\right)\mathbf {y} \rangle =0}
13805:
3957:{\displaystyle \langle (\mathbf {x} -P\mathbf {x} ),P\mathbf {y} \rangle =\langle P\mathbf {x} ,(\mathbf {y} -P\mathbf {y} )\rangle =0}
3376:{\displaystyle \langle P\mathbf {x} ,(\mathbf {y} -P\mathbf {y} )\rangle =\langle (\mathbf {x} -P\mathbf {x} ),P\mathbf {y} \rangle =0}
13817:
8106:
7988:
13861:
8485:
14194:
12933:. For Banach spaces, a one-dimensional subspace always has a closed complementary subspace. This is an immediate consequence of
9708:
14252:
9021:
7752:
7514:
5487:
5038:
4892:
13789:
13668:
Brust, J. J.; Marcia, R. F.; Petra, C. G. (2020), "Computationally
Efficient Decompositions of Oblique Projection Matrices",
9967:{\displaystyle \sigma _{i}={\begin{cases}{\sqrt {1+\gamma _{i}^{2}}}&1\leq i\leq k\\0&{\text{otherwise}}\end{cases}}}
6024:
5768:
5158:
2376:
10628:
6376:
2515:
10850:
3013:
191:. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on
17:
5804:
5712:
13760:
13644:
13584:
13536:
11663:
Many of the algebraic results discussed above survive the passage to this context. A given direct sum decomposition of
13134:
9277:
8040:
13482:
13420:
7799:
form a basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the matrix
3602:
8374:
462:
14242:
4561:
is complex-valued, the transpose in the above equation is replaced by a
Hermitian transpose). This operator leaves
4419:
3745:
14204:
14140:
13431:
10985:
6367:
8983:{\displaystyle P\mathbf {x} =\mathbf {x} _{1}=A\mathbf {w} =A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}\mathbf {x} }
2414:
13231:
2805:
9370:
6296:(i.e. the number of generators is greater than its dimension), the formula for the projection takes the form:
5967:{\displaystyle \mathbf {u} \left(\mathbf {u} ^{\mathsf {T}}\mathbf {u} \right)^{-1}\mathbf {u} ^{\mathsf {T}}}
598:
4223:
2962:
13339:
14291:
13982:
13854:
13237:
10912:
10663:
4613:
as the sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it,
3101:
The product of projections is not in general a projection, even if they are orthogonal. If two projections
13441:
14087:
12771:
2311:
1995:
1698:
13336:
one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that
6417:
3571:
1365:
13992:
13886:
13823:
13811:
13227:
13014:
11482:
10281:{\displaystyle \kappa (I-P)={\frac {\sigma _{1}}{1}}\geq {\frac {\sigma _{1}}{\sigma _{k}}}=\kappa (P)}
9212:
8993:
4185:
3126:
2348:
1938:
8721:
8456:
8322:
8250:
6194:{\textstyle P_{A}x=\operatorname {argmin} _{y\in \operatorname {range} (A)}\left\|x-y\right\|_{D}^{2}}
5432:
1887:
873:
703:
14301:
14232:
13881:
13304:
13266:
11925:
9344:
7699:
is the zero operator. The range and the kernel are complementary spaces, so the kernel has dimension
5762:
5284:
452:{\displaystyle \langle P\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,P\mathbf {y} \rangle }
13833:
10025:
9898:
9645:
7429:
3018:
230:
14224:
14107:
13712:
11636:, analytic questions, irrelevant in the finite-dimensional case, need to be considered. Assume now
4886:
1970:
653:
13634:
13574:
13526:
10722:
10694:
10555:
10404:
8554:
8434:
7946:
7406:), though not as frequently as orthogonal projections. Whereas calculating the fitted value of an
6299:
4594:
4568:
4542:
4459:
4203:
3529:
3507:
3240:
3218:
1916:
1865:
14296:
14270:
14199:
13977:
13847:
12934:
12162:
11706:
11576:
11414:
6293:
5148:{\displaystyle A={\begin{bmatrix}\mathbf {u} _{1}&\cdots &\mathbf {u} _{k}\end{bmatrix}}}
2316:
2146:
31:
110:
14034:
13967:
13957:
12041:
8695:
7584:
7407:
7387:
5535:
5012:
4413:
1672:
34:. For a concrete discussion of orthogonal projections in finite-dimensional linear spaces, see
13094:
12983:
12739:
11738:
11441:
9459:
14049:
14044:
14039:
13972:
13917:
13836:– a simple-to-follow tutorial explaining the different types of planar geometric projections.
13402:
13061:
12930:
12008:
11892:
11819:
11125:
11062:
10856:
10846:
10791:
9675:
7842:
7726:
7630:
7461:
7411:
5541:
5315:
4966:
4151:
3481:
3130:
2930:{\displaystyle (\lambda I-P)^{-1}={\frac {1}{\lambda }}I+{\frac {1}{\lambda (\lambda -1)}}P.}
2811:
1804:
1470:
1153:
1121:{\displaystyle P{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}x\\y\\0\end{bmatrix}}.}
902:
835:
538:
320:
262:
77:
11143:
1023:{\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.}
14059:
14024:
14011:
13902:
13258:
Projective elements of matrix algebras are used in the construction of certain K-groups in
13247:
12529:
11382:
11011:
9681:
9598:
9250:
6342:
6082:
5381:
2587:
1851:
756:
647:
188:
13599:
Banerjee, Sudipto (2004), "Revisiting
Spherical Trigonometry with Orthogonal Projectors",
13281:. Therefore, as one can imagine, projections are very often encountered in the context of
12561:
159:
is applied twice to any vector, it gives the same result as if it were applied once (i.e.
8:
14237:
14117:
14092:
13942:
13398:
13386:
13286:
11982:
11633:
11550:
11099:
10842:
10715:
10160:
9977:
9548:
9502:
9245:
9113:
7702:
7656:
7383:
3817:
2142:
1859:
781:
184:
69:
50:
8351:
8299:
13947:
13729:
13693:
13616:
13426:
13270:
13190:
13114:
12960:
12940:
12908:
12888:
12868:
12848:
12697:
12628:
12590:
12511:
12491:
12471:
12437:
12417:
12397:
12272:
12252:
12228:
12208:
12188:
12168:
12085:
11872:
11852:
11799:
11779:
11686:
11666:
11639:
11615:
11522:
11042:
10991:
10967:
10889:
10824:
10744:
10426:
10384:
10299:
10140:
9856:
9625:
9578:
9528:
9439:
9350:
9093:
8675:
8655:
8635:
8534:
8414:
8279:
8086:
7968:
7822:
7802:
7732:
7682:
7610:
7564:
7494:
7403:
7190:
6479:
6455:
6004:
5977:
5892:{\displaystyle \mathbf {u} ^{\mathsf {T}}\mathbf {u} =\left\|\mathbf {u} \right\|^{2},}
5692:
5672:
5567:
5464:
5412:
5361:
5341:
5321:
5272:{\displaystyle P_{A}=\sum _{i}\langle \mathbf {u} _{i},\cdot \rangle \mathbf {u} _{i}.}
4992:
4946:
4131:
4111:
4091:
3843:
3797:
3777:
3551:
3487:
3262:
3194:
3174:
3138:
3102:
3072:
2992:
2972:
2944:
2779:
2759:
2739:
2719:
2699:
2679:
2659:
2639:
2619:
2567:
2238:
2218:
2195:
2172:
2152:
2124:
2104:
2081:
2061:
2041:
1780:
1649:
1626:
1133:
787:
736:
574:
514:
374:
354:
326:
298:
210:
162:
142:
82:
14145:
14102:
14029:
13922:
13785:
13756:
13733:
13697:
13685:
13640:
13620:
13580:
13532:
13478:
13436:
13259:
13253:
13243:
12978:
10549:
9572:
5998:
5709:
into the underlying vector space but is no longer an isometry in general. The matrix
4940:
4885:
This formula can be generalized to orthogonal projections on a subspace of arbitrary
2213:
1855:
35:
8240:{\displaystyle P(\mathbf {x} _{2})=P(\mathbf {x} )-P^{2}(\mathbf {x} )=\mathbf {0} }
7421:
2505:{\displaystyle \mathbf {v} =\mathbf {x} -P\mathbf {x} =\left(I-P\right)\mathbf {x} }
14150:
14054:
13907:
13721:
13677:
13612:
13608:
13414:
13382:
13282:
13223:
10762:
10188:
5311:
4197:
3106:
1033:
192:
9244:. In general, if the vector space is over complex number field, one then uses the
4184:
The existence of an orthogonal projection onto a closed subspace follows from the
187:
unchanged. This definition of "projection" formalizes and generalizes the idea of
14209:
14002:
13962:
13952:
13290:
11036:
7410:
regression requires an orthogonal projection, calculating the fitted value of an
7382:
All these formulas also hold for complex inner product spaces, provided that the
2099:
54:
10886:, which splits into distinct linear factors. Thus there exists a basis in which
8878:{\displaystyle \mathbf {w} =(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}\mathbf {x} .}
14214:
14135:
13870:
13827:
13561:
Linear least squares (mathematics) § Properties of the least-squares estimators
13278:
13274:
10618:{\displaystyle \mathbf {y} =\operatorname {proj} _{V}\mathbf {y} +\mathbf {z} }
4587:, proving that it is indeed the orthogonal projection onto the line containing
3090:
1667:
697:
65:
30:"Orthogonal projection" redirects here. For the technical drawing concept, see
12384:{\displaystyle X=\operatorname {rg} (P)\oplus \ker(P)=\ker(1-P)\oplus \ker(P)}
11683:
into complementary subspaces still specifies a projection, and vice versa. If
14285:
14247:
14170:
14130:
14097:
14077:
12926:
12161:
However, in contrast to the finite-dimensional case, projections need not be
11147:
4483:
3212:
3164:
3160:
3156:
3109:
is false: the product of two non-commuting projections may be a projection.
807:
A projection matrix that is not an orthogonal projection matrix is called an
509:
348:
344:
316:
4658:{\displaystyle \mathbf {x} =\mathbf {x} _{\parallel }+\mathbf {x} _{\perp }}
4402:{\displaystyle \left\|P\mathbf {v} \right\|\leq \left\|\mathbf {v} \right\|}
14180:
14069:
14019:
13912:
11657:
11369:{\displaystyle \sigma _{1}\geq \sigma _{2}\geq \dots \geq \sigma _{k}>0}
6283:{\displaystyle P_{A}=A\left(A^{\mathsf {T}}DA\right)^{-1}A^{\mathsf {T}}D.}
3114:
104:
100:
14160:
14125:
14082:
13927:
11093:
10374:{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\dots ,\mathbf {u} _{p}}
10184:
5660:{\displaystyle P_{A}=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}.}
4879:
4479:
3860:
is orthogonal then it is self-adjoint, follows from the implication from
644:
12225:
is not continuous. In other words, the range of a continuous projection
8802:{\displaystyle B^{\mathsf {T}}(\mathbf {x} -A\mathbf {w} )=\mathbf {0} }
8625:{\displaystyle B^{\mathsf {T}}(\mathbf {x} -A\mathbf {w} )=\mathbf {0} }
4532:{\displaystyle P_{\mathbf {u} }=\mathbf {u} \mathbf {u} ^{\mathsf {T}}.}
41:
14189:
13932:
13725:
13681:
10316:
be a vector space (in this case a plane) spanned by orthogonal vectors
9846:{\displaystyle \gamma _{1}\geq \gamma _{2}\geq \ldots \geq \gamma _{k}}
6473:
4456:
A simple case occurs when the orthogonal projection is onto a line. If
2966:
2938:
1798:
1460:{\displaystyle P={\begin{bmatrix}0&0\\\alpha &1\end{bmatrix}}.}
816:
180:
13689:
6370:. This is just one of many ways to construct the projection operator.
495:. A projection on a Hilbert space that is not orthogonal is called an
13987:
13755:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
13639:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
13579:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
13531:, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC,
13216:
12151:{\displaystyle \operatorname {rg} (P)\oplus \operatorname {rg} (1-P)}
7934:{\displaystyle P=A\left(B^{\mathsf {T}}A\right)^{-1}B^{\mathsf {T}}.}
5584:
is the matrix with these vectors as columns, then the projection is:
2301:{\displaystyle \forall \mathbf {x} \in U:P\mathbf {x} =\mathbf {x} .}
1740:
730:
9202:{\displaystyle P=A\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}}
14155:
10768:
7180:{\displaystyle A^{\mathsf {T}}WB=A^{\mathsf {T}}W^{\mathsf {T}}B=0}
5761:
is a "normalizing factor" that recovers the norm. For example, the
1847:
1843:
13710:
Doković, D. Ž. (August 1991). "Unitary similarity of projectors".
13839:
13779:
11377:
7422:
A matrix representation formula for a nonzero projection operator
12394:
The converse holds also, with an additional assumption. Suppose
3840:
are orthogonal projections. The other direction, namely that if
14165:
11146:
operator. If the vector space is complex and equipped with an
8153:{\displaystyle \mathbf {x} _{2}=\mathbf {x} -P(\mathbf {x} ).}
8030:{\displaystyle \mathbf {x} =\mathbf {x} _{1}+\mathbf {x} _{2}}
4591:. A simple way to see this is to consider an arbitrary vector
27:
Idempotent linear transformation from a vector space to itself
12205:
is not closed in the norm topology, then the projection onto
10691:. There is a theorem in linear algebra that states that this
1405:
A simple example of a non-orthogonal (oblique) projection is
13215:
Projections (orthogonal and otherwise) play a major role in
8524:{\displaystyle B^{\mathsf {T}}\mathbf {x} _{2}=\mathbf {0} }
7418:
projection is an oblique projection, or just a projection.
2941:
of a projection. This implies that an orthogonal projection
10291:
10123:
9960:
2989:, there may be many projections whose range (or kernel) is
13210:
9787:{\displaystyle Q_{A}^{T}A(B^{T}A)^{-1}B^{T}Q_{A}^{\perp }}
4412:
For finite-dimensional complex or real vector spaces, the
13301:
More generally, given a map between normed vector spaces
12845:
The above argument makes use of the assumption that both
9080:{\displaystyle P=A(B^{\mathsf {T}}A)^{-1}B^{\mathsf {T}}}
7792:{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}}
7554:{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}
6292:
When the range space of the projection is generated by a
5527:{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}
5078:{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}
4932:{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}
4565:
invariant, and it annihilates all vectors orthogonal to
12905:, there need not exist a complementary closed subspace
11519:
corresponds to the maximal invariant subspace on which
3117:
commute if and only if their product is self-adjoint).
3069:, which factors into distinct linear factors, and thus
1397:
shows that the projection is an orthogonal projection.
13423:
to compute the projection onto an intersection of sets
11607:
11231:
11179:
10187:
of the oblique projections are the same. However, the
9525:
is also an oblique projection. The singular values of
8551:. Put these conditions together, and we find a vector
7249:
7225:
6948:
6879:
6854:
6800:
6762:
6718:
6688:
6634:
6581:
6547:
6522:
6385:
6112:
6072:{\displaystyle \langle x,y\rangle _{D}=y^{\dagger }Dx}
5794:{\displaystyle \mathbf {u} \mathbf {u} ^{\mathsf {T}}}
5103:
3484:. Using the self-adjoint and idempotent properties of
2404:{\displaystyle \mathbf {x} =\mathbf {u} +\mathbf {v} }
2033:
1573:
1534:
1498:
1423:
1319:
1280:
1244:
1205:
1087:
1051:
959:
13342:
13307:
13269:. Idempotents are used in classifying, for instance,
13193:
13137:
13117:
13097:
13064:
13017:
12986:
12963:
12943:
12911:
12891:
12871:
12851:
12774:
12742:
12700:
12631:
12593:
12564:
12514:
12494:
12474:
12440:
12420:
12400:
12299:
12275:
12255:
12231:
12211:
12191:
12171:
12108:
12088:
12044:
12011:
11985:
11928:
11895:
11875:
11855:
11822:
11802:
11782:
11741:
11709:
11689:
11669:
11642:
11618:
11579:
11553:
11525:
11485:
11444:
11417:
11385:
11317:
11167:
11128:
11102:
11065:
11045:
11014:
10994:
10970:
10915:
10892:
10859:
10827:
10794:
10747:
10725:
10697:
10666:
10653:{\displaystyle \operatorname {proj} _{V}\mathbf {y} }
10631:
10580:
10558:
10449:
10429:
10407:
10387:
10322:
10302:
10197:
10163:
10143:
10006:
9980:
9879:
9859:
9800:
9711:
9684:
9648:
9628:
9601:
9581:
9551:
9531:
9505:
9462:
9442:
9373:
9353:
9280:
9253:
9215:
9142:
9116:
9096:
9024:
8996:
8891:
8815:
8756:
8724:
8698:
8678:
8658:
8638:
8579:
8557:
8537:
8488:
8459:
8437:
8417:
8377:
8354:
8325:
8302:
8282:
8253:
8166:
8109:
8089:
8043:
7991:
7971:
7949:
7871:
7845:
7825:
7805:
7755:
7735:
7705:
7685:
7659:
7633:
7613:
7587:
7567:
7517:
7497:
7464:
7432:
7213:
7193:
7122:
6502:
6482:
6458:
6420:
6407:{\displaystyle {\begin{bmatrix}A&B\end{bmatrix}}}
6379:
6345:
6302:
6207:
6085:
6027:
6007:
5980:
5905:
5843:
5807:
5771:
5715:
5695:
5675:
5590:
5570:
5544:
5490:
5484:
The orthonormality condition can also be dropped. If
5467:
5435:
5415:
5384:
5364:
5344:
5324:
5287:
5206:
5161:
5091:
5041:
5015:
4995:
4969:
4949:
4895:
4671:
4619:
4597:
4571:
4545:
4491:
4462:
4422:
4364:
4232:
4206:
4154:
4134:
4114:
4094:
3970:
3866:
3846:
3820:
3800:
3780:
3748:
3644:
3605:
3574:
3554:
3532:
3510:
3490:
3389:
3285:
3265:
3243:
3221:
3197:
3177:
3141:
3075:
3021:
2995:
2975:
2947:
2846:
2814:
2782:
2762:
2742:
2722:
2702:
2682:
2662:
2642:
2622:
2590:
2570:
2518:
2450:
2417:
2379:
2351:
2319:
2261:
2241:
2221:
2198:
2175:
2155:
2127:
2107:
2084:
2064:
2044:
1998:
1973:
1941:
1919:
1890:
1868:
1807:
1783:
1701:
1675:
1652:
1629:
1479:
1411:
1368:
1189:
1156:
1136:
1042:
947:
905:
876:
838:
790:
759:
739:
706:
656:
601:
577:
541:
517:
465:
401:
377:
357:
329:
301:
265:
233:
213:
165:
145:
113:
85:
13277:
begins with considering characteristic functions of
2550:{\displaystyle \mathbf {u} \in U,\mathbf {v} \in V.}
9209:. By using this formula, one can easily check that
941:-plane. This function is represented by the matrix
13784:. Society for Industrial and Applied Mathematics.
13373:
13328:
13199:
13179:
13123:
13103:
13083:
13050:
12992:
12969:
12949:
12917:
12897:
12877:
12857:
12834:
12760:
12706:
12637:
12599:
12579:
12520:
12500:
12480:
12446:
12426:
12406:
12383:
12281:
12261:
12237:
12217:
12197:
12177:
12150:
12094:
12074:
12030:
11997:
11971:
11914:
11881:
11861:
11841:
11808:
11788:
11768:
11727:
11695:
11675:
11648:
11624:
11592:
11565:
11531:
11511:
11471:
11430:
11403:
11368:
11300:
11134:
11114:
11084:
11051:
11027:
11000:
10976:
10953:
10898:
10878:
10833:
10813:
10753:
10733:
10705:
10683:
10652:
10617:
10566:
10540:
10435:
10415:
10393:
10373:
10308:
10280:
10175:
10149:
10129:
9992:
9966:
9865:
9845:
9786:
9697:
9666:
9634:
9614:
9587:
9563:
9537:
9517:
9484:
9448:
9428:
9359:
9335:
9266:
9236:
9201:
9128:
9102:
9079:
9010:
8982:
8877:
8801:
8742:
8710:
8684:
8664:
8644:
8624:
8565:
8543:
8523:
8474:
8445:
8423:
8403:
8363:
8340:
8311:
8288:
8268:
8239:
8152:
8095:
8075:
8029:
7977:
7957:
7933:
7857:
7831:
7811:
7791:
7741:
7717:
7691:
7671:
7645:
7619:
7599:
7573:
7553:
7503:
7483:
7450:
7372:
7199:
7179:
7106:
6488:
6464:
6444:
6406:
6358:
6331:
6282:
6193:
6098:
6071:
6013:
5986:
5966:
5891:
5830:{\displaystyle \left\|\mathbf {u} \right\|\neq 1.}
5829:
5793:
5754:{\displaystyle \left(A^{\mathsf {T}}A\right)^{-1}}
5753:
5701:
5681:
5659:
5576:
5556:
5526:
5473:
5453:
5421:
5397:
5370:
5350:
5330:
5302:
5271:
5192:
5147:
5077:
5027:
5001:
4981:
4955:
4931:
4870:
4657:
4605:
4579:
4553:
4531:
4470:
4440:
4401:
4350:
4214:
4173:
4140:
4120:
4100:
4080:
3956:
3852:
3832:
3806:
3786:
3766:
3734:
3630:
3591:
3560:
3540:
3518:
3496:
3470:
3375:
3271:
3251:
3229:
3203:
3183:
3147:
3081:
3061:
3001:
2981:
2953:
2929:
2832:
2788:
2768:
2748:
2728:
2708:
2688:
2668:
2648:
2628:
2608:
2576:
2549:
2504:
2436:
2403:
2365:
2337:
2300:
2247:
2227:
2204:
2181:
2161:
2133:
2113:
2090:
2070:
2050:
2022:
1984:
1959:
1927:
1905:
1876:
1826:
1789:
1725:
1687:
1658:
1635:
1615:
1459:
1389:
1352:
1175:
1142:
1120:
1022:
929:
891:
862:
796:
772:
745:
721:
688:
635:
583:
560:
523:
487:
451:
383:
363:
335:
307:
284:
251:
219:
171:
151:
131:
91:
13819:Linear Algebra 15d: The Projection Transformation
13807:MIT Linear Algebra Lecture on Projection Matrices
13753:Linear Algebra and Matrix Analysis for Statistics
13636:Linear Algebra and Matrix Analysis for Statistics
13576:Linear Algebra and Matrix Analysis for Statistics
13528:Linear Algebra and Matrix Analysis for Statistics
10137:This implies that the largest singular values of
4482:on the line, then the projection is given by the
14283:
13670:SIAM Journal on Matrix Analysis and Applications
13667:
13180:{\displaystyle \ker(P)=\operatorname {rg} (I-P)}
12885:are closed. In general, given a closed subspace
9336:{\displaystyle P=A\left(A^{*}A\right)^{-1}A^{*}}
8076:{\displaystyle \mathbf {x} _{1}=P(\mathbf {x} )}
3480:A projection is orthogonal if and only if it is
2636:is also a projection as the image and kernel of
1735:
13768:
13389:. The case of an orthogonal projection is when
5378:into the underlying vector space. The range of
4191:
3631:{\displaystyle \mathbf {y} -P\mathbf {y} \in V}
832:For example, the function which maps the point
10574:can be written as an orthogonal sum such that
8404:{\displaystyle \mathbf {x} _{1}=A\mathbf {w} }
5997:In the general case, we can have an arbitrary
488:{\displaystyle \mathbf {x} ,\mathbf {y} \in V}
13855:
13468:
13466:
13417:, which is an example of a projection matrix.
4441:{\displaystyle \langle \cdot ,\cdot \rangle }
3767:{\displaystyle \langle \cdot ,\cdot \rangle }
13473:Horn, Roger A.; Johnson, Charles R. (2013).
10401:be a vector. One can define a projection of
6041:
6028:
5251:
5230:
4435:
4423:
4308:
4289:
4283:
4261:
4075:
4049:
4043:
4024:
4018:
3996:
3990:
3971:
3945:
3909:
3903:
3867:
3761:
3749:
3723:
3681:
3675:
3645:
3462:
3443:
3437:
3415:
3409:
3390:
3364:
3328:
3322:
3286:
3105:then their product is a projection, but the
2827:
2815:
446:
427:
421:
402:
13750:
13632:
13572:
13524:
13472:
9705:. Denote the singular values of the matrix
7116:If the orthogonal condition is enhanced to
2804:In infinite-dimensional vector spaces, the
13862:
13848:
13781:Matrix Analysis and Applied Linear Algebra
13463:
13091:, i.e. it is a projection. Boundedness of
11632:is a (not necessarily finite-dimensional)
10288:, and is therefore not necessarily equal.
7511:is not the zero operator. Let the vectors
2437:{\displaystyle \mathbf {u} =P\mathbf {x} }
1032:The action of this matrix on an arbitrary
12977:. By Hahn–Banach, there exists a bounded
10775:is being projected onto the vector space
9110:is an orthogonal projection, we can take
3120:
3096:
2560:The image and kernel of a projection are
2058:be a finite-dimensional vector space and
879:
13771:Linear Operators, Part I: General Theory
13751:Banerjee, Sudipto; Roy, Anindya (2014),
13633:Banerjee, Sudipto; Roy, Anindya (2014),
13598:
13573:Banerjee, Sudipto; Roy, Anindya (2014),
13525:Banerjee, Sudipto; Roy, Anindya (2014),
10767:
10548:where repeated indices are summed over (
10292:Finding projection with an inner product
9429:{\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}}
1739:
827:
636:{\displaystyle P^{2}=P=P^{\mathrm {T} }}
40:
13709:
13211:Applications and further considerations
4963:, with the assumption that the integer
4882:of parallel and perpendicular vectors.
819:of a projection matrix must be 0 or 1.
14:
14284:
14253:Comparison of linear algebra libraries
13401:, this is used in the definition of a
13374:{\displaystyle (\ker T)^{\perp }\to W}
13187:is a closed complementary subspace of
12929:this can always be done by taking the
10761:and is commonly used in areas such as
9228:
9193:
9164:
9071:
9043:
8969:
8941:
8861:
8833:
8763:
8731:
8586:
8495:
7922:
7893:
7393:
7350:
7318:
7296:
7264:
7162:
7150:
7129:
7094:
7065:
7041:
7012:
6973:
6957:
6917:
6888:
6825:
6809:
6743:
6727:
6659:
6643:
6606:
6590:
6427:
6268:
6236:
5958:
5925:
5852:
5785:
5728:
5648:
5619:
5445:
5294:
5193:{\displaystyle P_{A}=AA^{\mathsf {T}}}
5184:
4789:
4739:
4705:
4520:
1992:that is wholly contained in the image
535:if it is equal to its square, i.e. if
13843:
13834:Planar Geometric Projections Tutorial
13777:
13769:Dunford, N.; Schwartz, J. T. (1958).
13219:for certain linear algebra problems:
12528:is continuous. This follows from the
10954:{\displaystyle P=I_{r}\oplus 0_{d-r}}
10684:{\displaystyle {\hat {\mathbf {y} }}}
9853:. With this, the singular values for
3774:is the inner product associated with
3012:If a projection is nontrivial it has
1400:
937:is an orthogonal projection onto the
12434:. If there exists a closed subspace
11547:itself is orthogonal if and only if
7581:, and assemble these vectors in the
4222:in the vector space we have, by the
3171:is a projection for which the range
502:
12835:{\displaystyle P(x-y)=Px-Py=Px-y=0}
12005:is also a projection. The relation
11608:Projections on normed vector spaces
7207:non-singular, the following holds:
5534:is a (not necessarily orthonormal)
5155:. Then the projection is given by:
2034:Complementarity of image and kernel
2023:{\displaystyle P(B_{\mathbf {x} })}
1967:(with positive radius) centered on
1913:(with positive radius) centered on
1726:{\displaystyle P^{\mathrm {T} }=P.}
24:
13869:
13296:
11922:, then it is easily verified that
10783:
9495:
9343:. Recall that one can express the
6445:{\displaystyle A^{\mathsf {T}}B=0}
3592:{\displaystyle P\mathbf {x} \in U}
2262:
1708:
1390:{\displaystyle P^{\mathrm {T} }=P}
1375:
713:
627:
25:
14313:
13799:
13051:{\displaystyle P(x)=\varphi (x)u}
11776:is still a projection with range
11612:When the underlying vector space
11512:{\displaystyle I_{m}\oplus 0_{s}}
9237:{\displaystyle P=P^{\mathsf {T}}}
9011:{\displaystyle \mathbf {x} \in V}
7412:instrumental variables regression
2366:{\displaystyle \mathbf {x} \in W}
1960:{\displaystyle B_{P\mathbf {x} }}
14266:
14265:
14243:Basic Linear Algebra Subprograms
14001:
10727:
10699:
10671:
10646:
10611:
10603:
10582:
10560:
10528:
10513:
10498:
10484:
10475:
10464:
10409:
10361:
10340:
10325:
8998:
8976:
8922:
8905:
8896:
8868:
8817:
8795:
8784:
8773:
8750:is invertible. So the equation
8743:{\displaystyle B^{\mathsf {T}}A}
8618:
8607:
8596:
8559:
8517:
8503:
8475:{\displaystyle \mathbf {x} _{2}}
8462:
8439:
8397:
8380:
8341:{\displaystyle \mathbf {x} _{1}}
8328:
8269:{\displaystyle \mathbf {x} _{2}}
8256:
8233:
8222:
8198:
8175:
8140:
8126:
8112:
8066:
8046:
8017:
8002:
7993:
7951:
7779:
7758:
7541:
7520:
7414:requires an oblique projection.
5952:
5932:
5919:
5907:
5872:
5859:
5846:
5813:
5779:
5773:
5514:
5493:
5454:{\displaystyle AA^{\mathsf {T}}}
5256:
5235:
5127:
5108:
5065:
5044:
4919:
4898:
4858:
4849:
4841:
4818:
4797:
4783:
4761:
4747:
4733:
4727:
4713:
4699:
4693:
4685:
4678:
4645:
4630:
4621:
4599:
4573:
4547:
4514:
4508:
4498:
4464:
4391:
4374:
4340:
4323:
4304:
4296:
4279:
4268:
4243:
4208:
4071:
4053:
4039:
4031:
4014:
4003:
3986:
3975:
3938:
3927:
3916:
3899:
3885:
3874:
3719:
3685:
3671:
3660:
3652:
3618:
3607:
3579:
3534:
3512:
3458:
3450:
3433:
3422:
3405:
3394:
3360:
3346:
3335:
3315:
3304:
3293:
3245:
3223:
2965:. In general, the corresponding
2808:of a projection is contained in
2534:
2520:
2498:
2471:
2460:
2452:
2430:
2419:
2397:
2389:
2381:
2353:
2291:
2283:
2266:
2011:
1978:
1951:
1921:
1906:{\displaystyle B_{\mathbf {x} }}
1897:
1870:
892:{\displaystyle \mathbb {R} ^{3}}
722:{\displaystyle P^{\mathrm {T} }}
475:
467:
442:
431:
417:
409:
14141:Seven-dimensional cross product
13703:
13661:
13652:
13626:
13601:The College Mathematics Journal
13592:
13475:Matrix Analysis, second edition
13432:Least-squares spectral analysis
13329:{\displaystyle T\colon V\to W,}
11972:{\displaystyle (1-P)^{2}=(1-P)}
11735:, then the operator defined by
10821:on a vector space of dimension
7458:be a linear operator such that
5303:{\displaystyle A^{\mathsf {T}}}
2656:become the kernel and image of
13613:10.1080/07468342.2004.11922099
13566:
13553:
13544:
13518:
13509:
13500:
13491:
13477:. Cambridge University Press.
13454:
13421:Dykstra's projection algorithm
13365:
13356:
13343:
13317:
13174:
13162:
13150:
13144:
13042:
13036:
13027:
13021:
12790:
12778:
12378:
12372:
12360:
12348:
12336:
12330:
12318:
12312:
12145:
12133:
12121:
12115:
12069:
12057:
11966:
11954:
11942:
11929:
11757:
11745:
10713:is the smallest distance (the
10675:
10275:
10269:
10213:
10201:
9747:
9730:
9667:{\displaystyle Q_{A}^{\perp }}
9404:
9387:
9053:
9034:
8951:
8932:
8843:
8824:
8788:
8769:
8611:
8592:
8226:
8218:
8202:
8194:
8185:
8170:
8144:
8136:
8070:
8062:
7561:form a basis for the range of
7451:{\displaystyle P\colon V\to V}
7442:
6176:
6162:
6152:
6146:
5876:
5868:
5817:
5809:
4828:
4813:
4665:. Applying projection, we get
4395:
4387:
4379:
4366:
4344:
4336:
4328:
4315:
4248:
4235:
4196:An orthogonal projection is a
3942:
3923:
3889:
3870:
3350:
3331:
3319:
3300:
3062:{\displaystyle x^{2}-x=x(x-1)}
3056:
3044:
2915:
2903:
2863:
2847:
2373:may be decomposed uniquely as
2189:has the following properties:
2017:
2002:
1772:
1150:is indeed a projection, i.e.,
924:
906:
857:
839:
252:{\displaystyle P\colon V\to V}
243:
198:
13:
1:
13744:
13289:is generated by its complete
13246:form (the first step in many
11154:basis in which the matrix of
10183:are equal, and thus that the
8296:, which is the null space of
6414:is a non-singular matrix and
5974:onto the subspace spanned by
2963:positive semi-definite matrix
1985:{\displaystyle P\mathbf {x} }
1736:Properties and classification
689:{\displaystyle P^{2}=P=P^{*}}
49:is the orthogonal projection
13983:Eigenvalues and eigenvectors
13385:); in particular it must be
13238:Singular value decomposition
10734:{\displaystyle \mathbf {y} }
10706:{\displaystyle \mathbf {z} }
10567:{\displaystyle \mathbf {y} }
10416:{\displaystyle \mathbf {y} }
9974:and the singular values for
8566:{\displaystyle \mathbf {w} }
8446:{\displaystyle \mathbf {w} }
7958:{\displaystyle \mathbf {x} }
7729:of the kernel has dimension
6332:{\displaystyle P_{A}=AA^{+}}
5461:is the identity operator on
5358:is the isometry that embeds
4606:{\displaystyle \mathbf {x} }
4580:{\displaystyle \mathbf {u} }
4554:{\displaystyle \mathbf {u} }
4471:{\displaystyle \mathbf {u} }
4215:{\displaystyle \mathbf {v} }
4200:. This is because for every
4192:Properties and special cases
3541:{\displaystyle \mathbf {y} }
3519:{\displaystyle \mathbf {x} }
3252:{\displaystyle \mathbf {y} }
3230:{\displaystyle \mathbf {x} }
1928:{\displaystyle \mathbf {x} }
1877:{\displaystyle \mathbf {x} }
1846:, meaning that it maps each
1777:By definition, a projection
593:orthogonal projection matrix
7:
13408:
11728:{\displaystyle X=U\oplus V}
11593:{\displaystyle \sigma _{i}}
11431:{\displaystyle \sigma _{i}}
9622:be an orthonormal basis of
8692:by their construction, the
8319:In other words, the vector
6368:Moore–Penrose pseudoinverse
4451:
2799:
2338:{\displaystyle W=U\oplus V}
1862:. That is, for any vector
1837:
870:in three-dimensional space
822:
10:
14318:
13232:Gram–Schmidt decomposition
13228:Householder transformation
12842:, which proves the claim.
12165:in general. If a subspace
11600:-blocks correspond to the
11438:are uniquely determined.
8348:is in the column space of
6021:defining an inner product
5200:which can be rewritten as
4186:Hilbert projection theorem
3127:Hilbert projection theorem
3124:
351:can be used. A projection
132:{\displaystyle P\circ P=P}
29:
14261:
14223:
14179:
14116:
14068:
14010:
13999:
13895:
13877:
13814:, from MIT OpenCourseWare
13658:Meyer, equation (7.10.39)
12558:. One needs to show that
12269:gives a decomposition of
12075:{\displaystyle 1=P+(1-P)}
9456:has full column rank, so
8711:{\displaystyle k\times k}
7600:{\displaystyle n\times k}
6496:), the following holds:
5899:we obtain the projection
5035:matrix whose columns are
5028:{\displaystyle n\times k}
4878:by the properties of the
4224:Cauchy–Schwarz inequality
1688:{\displaystyle \alpha =0}
809:oblique projection matrix
13713:Aequationes Mathematicae
13550:Meyer, equation (5.13.3)
13515:Meyer, equation (5.13.4)
13447:
13381:be an isometry (compare
13267:characteristic functions
13104:{\displaystyle \varphi }
12993:{\displaystyle \varphi }
12761:{\displaystyle x-y\in V}
12414:is a closed subspace of
11816:. It is also clear that
11769:{\displaystyle P(u+v)=u}
11472:{\displaystyle 2k+s+m=d}
10660:is sometimes denoted as
9485:{\displaystyle P=AA^{+}}
5429:. It is also clear that
1748:is the projection along
1643:is indeed a projection.
13778:Meyer, Carl D. (2000).
13084:{\displaystyle P^{2}=P}
12289:into two complementary
12031:{\displaystyle P^{2}=P}
11915:{\displaystyle P^{2}=P}
11842:{\displaystyle P^{2}=P}
11135:{\displaystyle \oplus }
11085:{\displaystyle 0_{d-r}}
10879:{\displaystyle x^{2}-x}
10814:{\displaystyle P=P^{2}}
10191:satisfies the relation
9794:by the positive values
8531:by the construction of
7858:{\displaystyle k\geq 1}
7646:{\displaystyle k\geq 1}
7484:{\displaystyle P^{2}=P}
5801:is not a projection if
5557:{\displaystyle k\geq 1}
4982:{\displaystyle k\geq 1}
4416:can be substituted for
4174:{\displaystyle P=P^{*}}
2833:{\displaystyle \{0,1\}}
2676:and vice versa. We say
1842:Every projection is an
1827:{\displaystyle P^{2}=P}
1176:{\displaystyle P=P^{2}}
930:{\displaystyle (x,y,0)}
863:{\displaystyle (x,y,z)}
780:denotes the adjoint or
561:{\displaystyle P^{2}=P}
285:{\displaystyle P^{2}=P}
32:Orthographic projection
13968:Row and column vectors
13375:
13330:
13201:
13181:
13125:
13111:implies continuity of
13105:
13085:
13052:
12994:
12971:
12957:be the linear span of
12951:
12919:
12899:
12879:
12859:
12836:
12762:
12708:
12639:
12601:
12581:
12522:
12502:
12482:
12468:, then the projection
12448:
12428:
12408:
12385:
12283:
12263:
12239:
12219:
12199:
12179:
12152:
12096:
12076:
12032:
11999:
11973:
11916:
11883:
11863:
11843:
11810:
11790:
11770:
11729:
11697:
11677:
11650:
11626:
11594:
11567:
11533:
11513:
11473:
11432:
11405:
11370:
11302:
11136:
11116:
11086:
11053:
11029:
11002:
10978:
10955:
10900:
10880:
10835:
10815:
10780:
10755:
10735:
10707:
10685:
10654:
10619:
10568:
10542:
10437:
10417:
10395:
10375:
10310:
10282:
10177:
10151:
10131:
9994:
9968:
9867:
9847:
9788:
9699:
9668:
9636:
9616:
9589:
9571:can be computed by an
9565:
9539:
9519:
9486:
9450:
9430:
9361:
9337:
9268:
9238:
9203:
9136:, and it follows that
9130:
9104:
9081:
9012:
8984:
8879:
8803:
8744:
8712:
8686:
8666:
8646:
8626:
8567:
8545:
8525:
8476:
8447:
8425:
8405:
8365:
8342:
8313:
8290:
8270:
8241:
8154:
8097:
8077:
8031:
7979:
7959:
7935:
7859:
7833:
7819:. Then the projection
7813:
7793:
7743:
7725:. It follows that the
7719:
7693:
7673:
7647:
7621:
7601:
7575:
7555:
7505:
7485:
7452:
7408:ordinary least squares
7388:spherical trigonometry
7374:
7201:
7181:
7108:
6490:
6466:
6446:
6408:
6360:
6333:
6284:
6195:
6100:
6073:
6015:
5988:
5968:
5893:
5831:
5795:
5755:
5703:
5683:
5661:
5578:
5558:
5528:
5475:
5455:
5423:
5399:
5372:
5352:
5332:
5304:
5273:
5194:
5149:
5079:
5029:
5003:
4983:
4957:
4933:
4872:
4659:
4607:
4581:
4555:
4533:
4472:
4442:
4414:standard inner product
4403:
4352:
4216:
4175:
4142:
4122:
4102:
4082:
3958:
3854:
3834:
3808:
3788:
3768:
3736:
3632:
3593:
3562:
3542:
3520:
3498:
3472:
3377:
3273:
3253:
3231:
3205:
3185:
3159:and is complete (is a
3149:
3135:When the vector space
3121:Orthogonal projections
3097:Product of projections
3083:
3063:
3003:
2983:
2955:
2937:Only 0 or 1 can be an
2931:
2834:
2790:
2770:
2756:is a projection along
2750:
2730:
2710:
2696:is a projection along
2690:
2670:
2650:
2630:
2610:
2578:
2551:
2506:
2438:
2405:
2367:
2339:
2302:
2249:
2229:
2206:
2183:
2163:
2135:
2115:
2092:
2072:
2052:
2024:
1986:
1961:
1935:, there exists a ball
1929:
1907:
1878:
1854:to an open set in the
1828:
1791:
1769:
1727:
1689:
1660:
1637:
1617:
1461:
1391:
1354:
1177:
1144:
1122:
1024:
931:
893:
864:
798:
774:
747:
723:
690:
637:
585:
562:
525:
489:
453:
385:
365:
337:
309:
286:
253:
221:
173:
153:
133:
93:
61:
13973:Row and column spaces
13918:Scalar multiplication
13403:Riemannian submersion
13376:
13331:
13248:eigenvalue algorithms
13202:
13182:
13126:
13106:
13086:
13053:
12995:
12972:
12952:
12931:orthogonal complement
12920:
12900:
12880:
12860:
12837:
12763:
12709:
12640:
12602:
12582:
12523:
12503:
12483:
12449:
12429:
12409:
12386:
12284:
12264:
12240:
12220:
12200:
12180:
12153:
12097:
12077:
12033:
12000:
11974:
11917:
11884:
11864:
11844:
11811:
11791:
11771:
11730:
11698:
11678:
11651:
11627:
11595:
11568:
11534:
11514:
11474:
11433:
11411:and the real numbers
11406:
11404:{\displaystyle k,s,m}
11371:
11303:
11137:
11117:
11087:
11054:
11030:
11028:{\displaystyle I_{r}}
11003:
10979:
10956:
10901:
10881:
10847:diagonalizable matrix
10836:
10816:
10771:
10756:
10736:
10708:
10686:
10655:
10620:
10569:
10550:Einstein sum notation
10543:
10438:
10418:
10396:
10376:
10311:
10283:
10178:
10152:
10132:
9995:
9969:
9868:
9848:
9789:
9700:
9698:{\displaystyle Q_{A}}
9676:orthogonal complement
9669:
9637:
9617:
9615:{\displaystyle Q_{A}}
9590:
9566:
9540:
9520:
9487:
9451:
9431:
9362:
9345:Moore–Penrose inverse
9338:
9274:and has the formula
9269:
9267:{\displaystyle A^{*}}
9239:
9204:
9131:
9105:
9082:
9013:
8985:
8880:
8804:
8745:
8713:
8687:
8667:
8647:
8627:
8568:
8546:
8526:
8477:
8448:
8426:
8406:
8366:
8343:
8314:
8291:
8271:
8242:
8155:
8098:
8078:
8032:
7980:
7960:
7936:
7860:
7834:
7814:
7794:
7744:
7727:orthogonal complement
7720:
7694:
7674:
7648:
7622:
7602:
7576:
7556:
7506:
7486:
7453:
7375:
7202:
7182:
7109:
6491:
6467:
6447:
6409:
6361:
6359:{\displaystyle A^{+}}
6334:
6285:
6196:
6101:
6099:{\displaystyle P_{A}}
6079:, and the projection
6074:
6016:
5989:
5969:
5894:
5832:
5796:
5756:
5704:
5684:
5662:
5579:
5559:
5529:
5476:
5456:
5424:
5400:
5398:{\displaystyle P_{A}}
5373:
5353:
5333:
5316:orthogonal complement
5314:that vanishes on the
5305:
5274:
5195:
5150:
5080:
5030:
5004:
4984:
4958:
4934:
4873:
4660:
4608:
4582:
4556:
4534:
4473:
4443:
4404:
4353:
4217:
4176:
4143:
4123:
4103:
4083:
3959:
3855:
3835:
3809:
3789:
3769:
3737:
3633:
3594:
3563:
3543:
3521:
3499:
3473:
3378:
3274:
3254:
3232:
3206:
3186:
3169:orthogonal projection
3150:
3131:Complemented subspace
3084:
3064:
3004:
2984:
2956:
2932:
2835:
2791:
2771:
2751:
2731:
2711:
2691:
2671:
2651:
2631:
2611:
2609:{\displaystyle Q=I-P}
2579:
2552:
2507:
2439:
2406:
2368:
2340:
2303:
2250:
2230:
2207:
2184:
2164:
2136:
2116:
2093:
2073:
2053:
2025:
1987:
1962:
1930:
1908:
1879:
1829:
1792:
1743:
1728:
1690:
1661:
1638:
1618:
1471:matrix multiplication
1462:
1392:
1355:
1178:
1145:
1123:
1025:
932:
894:
865:
828:Orthogonal projection
799:
775:
773:{\displaystyle P^{*}}
748:
724:
691:
638:
586:
563:
526:
490:
454:
393:orthogonal projection
386:
366:
338:
310:
287:
254:
227:is a linear operator
222:
174:
154:
134:
94:
78:linear transformation
44:
14108:Gram–Schmidt process
14060:Gaussian elimination
13340:
13305:
13191:
13135:
13115:
13095:
13062:
13015:
12984:
12961:
12941:
12909:
12889:
12869:
12849:
12772:
12740:
12698:
12629:
12591:
12580:{\displaystyle Px=y}
12562:
12530:closed graph theorem
12512:
12492:
12472:
12438:
12418:
12398:
12297:
12273:
12253:
12229:
12209:
12189:
12169:
12106:
12086:
12042:
12009:
11983:
11926:
11893:
11873:
11853:
11820:
11800:
11780:
11739:
11707:
11687:
11667:
11640:
11616:
11577:
11551:
11543:projection (so that
11523:
11483:
11442:
11415:
11383:
11315:
11165:
11126:
11100:
11063:
11043:
11012:
10992:
10968:
10913:
10890:
10857:
10825:
10792:
10745:
10723:
10695:
10664:
10629:
10578:
10556:
10447:
10427:
10405:
10385:
10320:
10300:
10195:
10161:
10141:
10004:
9978:
9877:
9857:
9798:
9709:
9682:
9646:
9626:
9599:
9579:
9549:
9529:
9503:
9460:
9440:
9371:
9351:
9278:
9251:
9213:
9140:
9114:
9094:
9022:
8994:
8889:
8813:
8754:
8722:
8696:
8676:
8656:
8636:
8577:
8555:
8535:
8486:
8457:
8435:
8415:
8375:
8352:
8323:
8300:
8280:
8276:is in the kernel of
8251:
8164:
8107:
8087:
8041:
7989:
7969:
7965:in the vector space
7947:
7869:
7843:
7839:(with the condition
7823:
7803:
7753:
7733:
7703:
7683:
7657:
7631:
7611:
7585:
7565:
7515:
7495:
7462:
7430:
7211:
7191:
7120:
6500:
6480:
6456:
6418:
6377:
6343:
6300:
6205:
6110:
6083:
6025:
6005:
5978:
5903:
5841:
5805:
5769:
5713:
5693:
5673:
5588:
5568:
5542:
5488:
5465:
5433:
5413:
5382:
5362:
5342:
5322:
5285:
5204:
5159:
5089:
5039:
5013:
4993:
4967:
4947:
4893:
4669:
4617:
4595:
4569:
4543:
4489:
4460:
4420:
4362:
4230:
4204:
4152:
4132:
4112:
4092:
3968:
3864:
3844:
3818:
3798:
3778:
3746:
3642:
3603:
3572:
3552:
3530:
3508:
3488:
3387:
3283:
3263:
3241:
3219:
3213:orthogonal subspaces
3195:
3175:
3139:
3073:
3019:
2993:
2973:
2945:
2844:
2812:
2780:
2760:
2740:
2720:
2700:
2680:
2660:
2640:
2620:
2588:
2568:
2516:
2448:
2415:
2377:
2349:
2317:
2259:
2239:
2219:
2196:
2173:
2153:
2125:
2105:
2082:
2062:
2042:
1996:
1971:
1939:
1917:
1888:
1866:
1805:
1781:
1699:
1673:
1650:
1627:
1477:
1409:
1366:
1187:
1154:
1134:
1040:
945:
903:
874:
836:
788:
757:
737:
704:
654:
599:
575:
539:
515:
463:
399:
375:
355:
327:
299:
263:
231:
211:
189:graphical projection
163:
143:
139:. That is, whenever
111:
83:
14292:Functional analysis
14238:Numerical stability
14118:Multilinear algebra
14093:Inner product space
13943:Linear independence
13442:Properties of trace
13399:Riemannian geometry
13287:von Neumann algebra
13285:. In particular, a
13271:semisimple algebras
12935:Hahn–Banach theorem
11998:{\displaystyle 1-P}
11634:normed vector space
11566:{\displaystyle k=0}
11150:, then there is an
11115:{\displaystyle d-r}
10716:orthogonal distance
10176:{\displaystyle I-P}
10050:
9993:{\displaystyle I-P}
9923:
9783:
9726:
9663:
9564:{\displaystyle I-P}
9518:{\displaystyle I-P}
9246:Hermitian transpose
9129:{\displaystyle A=B}
8083:is in the image of
7985:, we can decompose
7718:{\displaystyle n-k}
7672:{\displaystyle k=0}
7400:oblique projections
7394:Oblique projections
7384:conjugate transpose
6190:
3833:{\displaystyle I-P}
2736:(kernel/image) and
2169:respectively. Then
2078:be a projection on
1744:The transformation
782:Hermitian transpose
650:, and respectively
371:on a Hilbert space
70:functional analysis
45:The transformation
18:Projection operator
13948:Linear combination
13726:10.1007/BF01818492
13682:10.1137/19M1288115
13427:Invariant subspace
13371:
13326:
13197:
13177:
13121:
13101:
13081:
13048:
12990:
12967:
12947:
12915:
12895:
12875:
12855:
12832:
12758:
12704:
12635:
12597:
12577:
12518:
12498:
12478:
12444:
12424:
12404:
12381:
12279:
12259:
12235:
12215:
12195:
12175:
12148:
12102:is the direct sum
12092:
12072:
12028:
11995:
11979:. In other words,
11969:
11912:
11879:
11859:
11839:
11806:
11786:
11766:
11725:
11703:is the direct sum
11693:
11673:
11646:
11622:
11590:
11563:
11529:
11509:
11469:
11428:
11401:
11366:
11298:
11263:
11211:
11132:
11112:
11082:
11049:
11025:
10998:
10974:
10951:
10896:
10876:
10851:minimal polynomial
10831:
10811:
10781:
10751:
10731:
10703:
10681:
10650:
10615:
10564:
10538:
10433:
10413:
10391:
10371:
10306:
10278:
10173:
10147:
10127:
10122:
10036:
9990:
9964:
9959:
9909:
9863:
9843:
9784:
9769:
9712:
9695:
9664:
9649:
9632:
9612:
9585:
9561:
9535:
9515:
9482:
9446:
9426:
9357:
9333:
9264:
9234:
9199:
9126:
9100:
9077:
9008:
8980:
8875:
8799:
8740:
8708:
8682:
8662:
8642:
8622:
8563:
8541:
8521:
8472:
8443:
8421:
8401:
8364:{\displaystyle A,}
8361:
8338:
8312:{\displaystyle A.}
8309:
8286:
8266:
8237:
8150:
8093:
8073:
8027:
7975:
7955:
7931:
7855:
7829:
7809:
7789:
7739:
7715:
7689:
7669:
7643:
7617:
7597:
7571:
7551:
7501:
7481:
7448:
7404:oblique projection
7370:
7358:
7238:
7197:
7177:
7104:
7102:
6981:
6928:
6867:
6833:
6775:
6751:
6701:
6667:
6614:
6560:
6535:
6486:
6462:
6442:
6404:
6398:
6356:
6329:
6280:
6191:
6160:
6096:
6069:
6011:
5984:
5964:
5889:
5837:After dividing by
5827:
5791:
5751:
5699:
5679:
5657:
5574:
5554:
5524:
5471:
5451:
5419:
5395:
5368:
5348:
5328:
5300:
5269:
5229:
5190:
5145:
5139:
5075:
5025:
4999:
4979:
4953:
4929:
4868:
4655:
4603:
4577:
4551:
4529:
4468:
4438:
4399:
4348:
4212:
4171:
4138:
4118:
4098:
4078:
3954:
3850:
3830:
3804:
3784:
3764:
3732:
3628:
3589:
3558:
3538:
3516:
3494:
3468:
3373:
3269:
3249:
3227:
3215:. Thus, for every
3201:
3181:
3145:
3079:
3059:
3014:minimal polynomial
2999:
2979:
2951:
2927:
2830:
2786:
2766:
2746:
2726:
2706:
2686:
2666:
2646:
2626:
2606:
2574:
2547:
2502:
2434:
2401:
2363:
2335:
2298:
2245:
2225:
2202:
2179:
2159:
2131:
2111:
2088:
2068:
2048:
2020:
1982:
1957:
1925:
1903:
1874:
1824:
1787:
1770:
1764:and the kernel is
1723:
1695:because only then
1685:
1656:
1633:
1613:
1598:
1559:
1523:
1457:
1448:
1401:Oblique projection
1387:
1350:
1341:
1302:
1266:
1227:
1173:
1140:
1118:
1109:
1073:
1020:
1011:
927:
889:
860:
794:
770:
743:
719:
686:
633:
581:
558:
521:
497:oblique projection
485:
449:
381:
361:
333:
305:
282:
249:
217:
207:on a vector space
169:
149:
129:
89:
62:
14279:
14278:
14146:Geometric algebra
14103:Kronecker product
13938:Linear projection
13923:Vector projection
13791:978-0-89871-454-8
13460:Meyer, pp 386+387
13437:Orthogonalization
13393:is a subspace of
13283:operator algebras
13260:Operator K-theory
13254:Linear regression
13200:{\displaystyle U}
13124:{\displaystyle P}
12979:linear functional
12970:{\displaystyle u}
12950:{\displaystyle U}
12918:{\displaystyle V}
12898:{\displaystyle U}
12878:{\displaystyle V}
12858:{\displaystyle U}
12707:{\displaystyle V}
12638:{\displaystyle U}
12600:{\displaystyle U}
12521:{\displaystyle V}
12501:{\displaystyle U}
12481:{\displaystyle P}
12447:{\displaystyle V}
12427:{\displaystyle X}
12407:{\displaystyle U}
12282:{\displaystyle X}
12262:{\displaystyle P}
12238:{\displaystyle P}
12218:{\displaystyle U}
12198:{\displaystyle X}
12178:{\displaystyle U}
12095:{\displaystyle X}
11882:{\displaystyle X}
11869:is projection on
11862:{\displaystyle P}
11849:. Conversely, if
11809:{\displaystyle V}
11789:{\displaystyle U}
11696:{\displaystyle X}
11676:{\displaystyle X}
11649:{\displaystyle X}
11625:{\displaystyle X}
11532:{\displaystyle P}
11052:{\displaystyle r}
11001:{\displaystyle P}
10977:{\displaystyle r}
10899:{\displaystyle P}
10834:{\displaystyle d}
10754:{\displaystyle V}
10678:
10524:
10436:{\displaystyle V}
10394:{\displaystyle y}
10309:{\displaystyle V}
10261:
10234:
10150:{\displaystyle P}
10118:
10051:
9955:
9924:
9866:{\displaystyle P}
9635:{\displaystyle A}
9588:{\displaystyle A}
9573:orthonormal basis
9538:{\displaystyle P}
9449:{\displaystyle A}
9360:{\displaystyle A}
9103:{\displaystyle P}
9090:In the case that
8809:gives the vector
8685:{\displaystyle k}
8672:are of full rank
8665:{\displaystyle B}
8645:{\displaystyle A}
8632:. Since matrices
8544:{\displaystyle B}
8431:dimension vector
8424:{\displaystyle k}
8289:{\displaystyle P}
8096:{\displaystyle P}
7978:{\displaystyle V}
7832:{\displaystyle P}
7812:{\displaystyle B}
7742:{\displaystyle k}
7692:{\displaystyle P}
7620:{\displaystyle A}
7574:{\displaystyle P}
7504:{\displaystyle P}
7200:{\displaystyle W}
6489:{\displaystyle A}
6465:{\displaystyle B}
6014:{\displaystyle D}
5999:positive definite
5987:{\displaystyle u}
5702:{\displaystyle U}
5682:{\displaystyle A}
5577:{\displaystyle A}
5474:{\displaystyle U}
5422:{\displaystyle A}
5405:is therefore the
5371:{\displaystyle U}
5351:{\displaystyle A}
5331:{\displaystyle U}
5220:
5002:{\displaystyle A}
4956:{\displaystyle U}
4941:orthonormal basis
4141:{\displaystyle W}
4121:{\displaystyle y}
4101:{\displaystyle x}
3853:{\displaystyle P}
3807:{\displaystyle P}
3787:{\displaystyle W}
3561:{\displaystyle W}
3497:{\displaystyle P}
3272:{\displaystyle W}
3204:{\displaystyle V}
3184:{\displaystyle U}
3163:) the concept of
3148:{\displaystyle W}
3082:{\displaystyle P}
3002:{\displaystyle V}
2982:{\displaystyle V}
2954:{\displaystyle P}
2919:
2886:
2789:{\displaystyle V}
2769:{\displaystyle U}
2749:{\displaystyle Q}
2729:{\displaystyle U}
2709:{\displaystyle V}
2689:{\displaystyle P}
2669:{\displaystyle Q}
2649:{\displaystyle P}
2629:{\displaystyle Q}
2577:{\displaystyle P}
2248:{\displaystyle U}
2228:{\displaystyle I}
2214:identity operator
2205:{\displaystyle P}
2182:{\displaystyle P}
2162:{\displaystyle P}
2134:{\displaystyle V}
2114:{\displaystyle U}
2091:{\displaystyle W}
2071:{\displaystyle P}
2051:{\displaystyle W}
1856:subspace topology
1790:{\displaystyle P}
1659:{\displaystyle P}
1636:{\displaystyle P}
1143:{\displaystyle P}
797:{\displaystyle P}
746:{\displaystyle P}
584:{\displaystyle P}
533:projection matrix
524:{\displaystyle P}
503:Projection matrix
384:{\displaystyle V}
364:{\displaystyle P}
347:, the concept of
336:{\displaystyle V}
308:{\displaystyle V}
220:{\displaystyle V}
183:). It leaves its
172:{\displaystyle P}
152:{\displaystyle P}
92:{\displaystyle P}
36:Vector projection
16:(Redirected from
14309:
14302:Linear operators
14269:
14268:
14151:Exterior algebra
14088:Hadamard product
14005:
13993:Linear equations
13864:
13857:
13850:
13841:
13840:
13820:
13808:
13795:
13774:
13765:
13738:
13737:
13707:
13701:
13700:
13665:
13659:
13656:
13650:
13649:
13630:
13624:
13623:
13596:
13590:
13589:
13570:
13564:
13557:
13551:
13548:
13542:
13541:
13522:
13516:
13513:
13507:
13504:
13498:
13495:
13489:
13488:
13470:
13461:
13458:
13415:Centering matrix
13383:Partial isometry
13380:
13378:
13377:
13372:
13364:
13363:
13335:
13333:
13332:
13327:
13293:of projections.
13224:QR decomposition
13206:
13204:
13203:
13198:
13186:
13184:
13183:
13178:
13130:
13128:
13127:
13122:
13110:
13108:
13107:
13102:
13090:
13088:
13087:
13082:
13074:
13073:
13057:
13055:
13054:
13049:
13010:
12999:
12997:
12996:
12991:
12976:
12974:
12973:
12968:
12956:
12954:
12953:
12948:
12924:
12922:
12921:
12916:
12904:
12902:
12901:
12896:
12884:
12882:
12881:
12876:
12864:
12862:
12861:
12856:
12841:
12839:
12838:
12833:
12767:
12765:
12764:
12759:
12735:
12713:
12711:
12710:
12705:
12693:
12654:
12644:
12642:
12641:
12636:
12620:
12606:
12604:
12603:
12598:
12586:
12584:
12583:
12578:
12557:
12544:
12527:
12525:
12524:
12519:
12507:
12505:
12504:
12499:
12487:
12485:
12484:
12479:
12467:
12453:
12451:
12450:
12445:
12433:
12431:
12430:
12425:
12413:
12411:
12410:
12405:
12390:
12388:
12387:
12382:
12288:
12286:
12285:
12280:
12268:
12266:
12265:
12260:
12244:
12242:
12241:
12236:
12224:
12222:
12221:
12216:
12204:
12202:
12201:
12196:
12184:
12182:
12181:
12176:
12157:
12155:
12154:
12149:
12101:
12099:
12098:
12093:
12081:
12079:
12078:
12073:
12037:
12035:
12034:
12029:
12021:
12020:
12004:
12002:
12001:
11996:
11978:
11976:
11975:
11970:
11950:
11949:
11921:
11919:
11918:
11913:
11905:
11904:
11888:
11886:
11885:
11880:
11868:
11866:
11865:
11860:
11848:
11846:
11845:
11840:
11832:
11831:
11815:
11813:
11812:
11807:
11795:
11793:
11792:
11787:
11775:
11773:
11772:
11767:
11734:
11732:
11731:
11726:
11702:
11700:
11699:
11694:
11682:
11680:
11679:
11674:
11655:
11653:
11652:
11647:
11631:
11629:
11628:
11623:
11599:
11597:
11596:
11591:
11589:
11588:
11572:
11570:
11569:
11564:
11538:
11536:
11535:
11530:
11518:
11516:
11515:
11510:
11508:
11507:
11495:
11494:
11478:
11476:
11475:
11470:
11437:
11435:
11434:
11429:
11427:
11426:
11410:
11408:
11407:
11402:
11375:
11373:
11372:
11367:
11359:
11358:
11340:
11339:
11327:
11326:
11307:
11305:
11304:
11299:
11294:
11293:
11281:
11280:
11268:
11267:
11248:
11247:
11216:
11215:
11196:
11195:
11141:
11139:
11138:
11133:
11121:
11119:
11118:
11113:
11091:
11089:
11088:
11083:
11081:
11080:
11058:
11056:
11055:
11050:
11034:
11032:
11031:
11026:
11024:
11023:
11007:
11005:
11004:
10999:
10983:
10981:
10980:
10975:
10960:
10958:
10957:
10952:
10950:
10949:
10931:
10930:
10905:
10903:
10902:
10897:
10885:
10883:
10882:
10877:
10869:
10868:
10840:
10838:
10837:
10832:
10820:
10818:
10817:
10812:
10810:
10809:
10763:machine learning
10760:
10758:
10757:
10752:
10740:
10738:
10737:
10732:
10730:
10712:
10710:
10709:
10704:
10702:
10690:
10688:
10687:
10682:
10680:
10679:
10674:
10669:
10659:
10657:
10656:
10651:
10649:
10641:
10640:
10624:
10622:
10621:
10616:
10614:
10606:
10598:
10597:
10585:
10573:
10571:
10570:
10565:
10563:
10547:
10545:
10544:
10539:
10537:
10536:
10531:
10525:
10523:
10522:
10521:
10516:
10507:
10506:
10501:
10494:
10493:
10492:
10487:
10478:
10472:
10467:
10459:
10458:
10442:
10440:
10439:
10434:
10422:
10420:
10419:
10414:
10412:
10400:
10398:
10397:
10392:
10380:
10378:
10377:
10372:
10370:
10369:
10364:
10349:
10348:
10343:
10334:
10333:
10328:
10315:
10313:
10312:
10307:
10287:
10285:
10284:
10279:
10262:
10260:
10259:
10250:
10249:
10240:
10235:
10230:
10229:
10220:
10189:condition number
10182:
10180:
10179:
10174:
10156:
10154:
10153:
10148:
10136:
10134:
10133:
10128:
10126:
10125:
10119:
10116:
10052:
10049:
10044:
10029:
10016:
10015:
9999:
9997:
9996:
9991:
9973:
9971:
9970:
9965:
9963:
9962:
9956:
9953:
9925:
9922:
9917:
9902:
9889:
9888:
9872:
9870:
9869:
9864:
9852:
9850:
9849:
9844:
9842:
9841:
9823:
9822:
9810:
9809:
9793:
9791:
9790:
9785:
9782:
9777:
9768:
9767:
9758:
9757:
9742:
9741:
9725:
9720:
9704:
9702:
9701:
9696:
9694:
9693:
9673:
9671:
9670:
9665:
9662:
9657:
9641:
9639:
9638:
9633:
9621:
9619:
9618:
9613:
9611:
9610:
9594:
9592:
9591:
9586:
9570:
9568:
9567:
9562:
9544:
9542:
9541:
9536:
9524:
9522:
9521:
9516:
9491:
9489:
9488:
9483:
9481:
9480:
9455:
9453:
9452:
9447:
9435:
9433:
9432:
9427:
9425:
9424:
9415:
9414:
9399:
9398:
9383:
9382:
9366:
9364:
9363:
9358:
9342:
9340:
9339:
9334:
9332:
9331:
9322:
9321:
9313:
9309:
9305:
9304:
9273:
9271:
9270:
9265:
9263:
9262:
9243:
9241:
9240:
9235:
9233:
9232:
9231:
9208:
9206:
9205:
9200:
9198:
9197:
9196:
9186:
9185:
9177:
9173:
9169:
9168:
9167:
9135:
9133:
9132:
9127:
9109:
9107:
9106:
9101:
9086:
9084:
9083:
9078:
9076:
9075:
9074:
9064:
9063:
9048:
9047:
9046:
9017:
9015:
9014:
9009:
9001:
8989:
8987:
8986:
8981:
8979:
8974:
8973:
8972:
8962:
8961:
8946:
8945:
8944:
8925:
8914:
8913:
8908:
8899:
8884:
8882:
8881:
8876:
8871:
8866:
8865:
8864:
8854:
8853:
8838:
8837:
8836:
8820:
8808:
8806:
8805:
8800:
8798:
8787:
8776:
8768:
8767:
8766:
8749:
8747:
8746:
8741:
8736:
8735:
8734:
8717:
8715:
8714:
8709:
8691:
8689:
8688:
8683:
8671:
8669:
8668:
8663:
8651:
8649:
8648:
8643:
8631:
8629:
8628:
8623:
8621:
8610:
8599:
8591:
8590:
8589:
8572:
8570:
8569:
8564:
8562:
8550:
8548:
8547:
8542:
8530:
8528:
8527:
8522:
8520:
8512:
8511:
8506:
8500:
8499:
8498:
8481:
8479:
8478:
8473:
8471:
8470:
8465:
8452:
8450:
8449:
8444:
8442:
8430:
8428:
8427:
8422:
8410:
8408:
8407:
8402:
8400:
8389:
8388:
8383:
8370:
8368:
8367:
8362:
8347:
8345:
8344:
8339:
8337:
8336:
8331:
8318:
8316:
8315:
8310:
8295:
8293:
8292:
8287:
8275:
8273:
8272:
8267:
8265:
8264:
8259:
8246:
8244:
8243:
8238:
8236:
8225:
8217:
8216:
8201:
8184:
8183:
8178:
8159:
8157:
8156:
8151:
8143:
8129:
8121:
8120:
8115:
8102:
8100:
8099:
8094:
8082:
8080:
8079:
8074:
8069:
8055:
8054:
8049:
8036:
8034:
8033:
8028:
8026:
8025:
8020:
8011:
8010:
8005:
7996:
7984:
7982:
7981:
7976:
7964:
7962:
7961:
7956:
7954:
7940:
7938:
7937:
7932:
7927:
7926:
7925:
7915:
7914:
7906:
7902:
7898:
7897:
7896:
7864:
7862:
7861:
7856:
7838:
7836:
7835:
7830:
7818:
7816:
7815:
7810:
7798:
7796:
7795:
7790:
7788:
7787:
7782:
7767:
7766:
7761:
7748:
7746:
7745:
7740:
7724:
7722:
7721:
7716:
7698:
7696:
7695:
7690:
7678:
7676:
7675:
7670:
7652:
7650:
7649:
7644:
7626:
7624:
7623:
7618:
7606:
7604:
7603:
7598:
7580:
7578:
7577:
7572:
7560:
7558:
7557:
7552:
7550:
7549:
7544:
7529:
7528:
7523:
7510:
7508:
7507:
7502:
7491:and assume that
7490:
7488:
7487:
7482:
7474:
7473:
7457:
7455:
7454:
7449:
7379:
7377:
7376:
7371:
7363:
7362:
7355:
7354:
7353:
7343:
7342:
7334:
7330:
7323:
7322:
7321:
7301:
7300:
7299:
7289:
7288:
7280:
7276:
7269:
7268:
7267:
7243:
7242:
7206:
7204:
7203:
7198:
7186:
7184:
7183:
7178:
7167:
7166:
7165:
7155:
7154:
7153:
7134:
7133:
7132:
7113:
7111:
7110:
7105:
7103:
7099:
7098:
7097:
7087:
7086:
7078:
7074:
7070:
7069:
7068:
7046:
7045:
7044:
7034:
7033:
7025:
7021:
7017:
7016:
7015:
6990:
6986:
6985:
6978:
6977:
6976:
6962:
6961:
6960:
6942:
6941:
6933:
6932:
6922:
6921:
6920:
6893:
6892:
6891:
6872:
6871:
6842:
6838:
6837:
6830:
6829:
6828:
6814:
6813:
6812:
6794:
6793:
6785:
6781:
6780:
6779:
6756:
6755:
6748:
6747:
6746:
6732:
6731:
6730:
6706:
6705:
6676:
6672:
6671:
6664:
6663:
6662:
6648:
6647:
6646:
6628:
6627:
6619:
6618:
6611:
6610:
6609:
6595:
6594:
6593:
6574:
6573:
6565:
6564:
6540:
6539:
6495:
6493:
6492:
6487:
6471:
6469:
6468:
6463:
6451:
6449:
6448:
6443:
6432:
6431:
6430:
6413:
6411:
6410:
6405:
6403:
6402:
6365:
6363:
6362:
6357:
6355:
6354:
6338:
6336:
6335:
6330:
6328:
6327:
6312:
6311:
6289:
6287:
6286:
6281:
6273:
6272:
6271:
6261:
6260:
6252:
6248:
6241:
6240:
6239:
6217:
6216:
6200:
6198:
6197:
6192:
6189:
6184:
6179:
6175:
6156:
6155:
6122:
6121:
6105:
6103:
6102:
6097:
6095:
6094:
6078:
6076:
6075:
6070:
6062:
6061:
6049:
6048:
6020:
6018:
6017:
6012:
5993:
5991:
5990:
5985:
5973:
5971:
5970:
5965:
5963:
5962:
5961:
5955:
5949:
5948:
5940:
5936:
5935:
5930:
5929:
5928:
5922:
5910:
5898:
5896:
5895:
5890:
5885:
5884:
5879:
5875:
5862:
5857:
5856:
5855:
5849:
5836:
5834:
5833:
5828:
5820:
5816:
5800:
5798:
5797:
5792:
5790:
5789:
5788:
5782:
5776:
5760:
5758:
5757:
5752:
5750:
5749:
5741:
5737:
5733:
5732:
5731:
5708:
5706:
5705:
5700:
5688:
5686:
5685:
5680:
5666:
5664:
5663:
5658:
5653:
5652:
5651:
5641:
5640:
5632:
5628:
5624:
5623:
5622:
5600:
5599:
5583:
5581:
5580:
5575:
5563:
5561:
5560:
5555:
5533:
5531:
5530:
5525:
5523:
5522:
5517:
5502:
5501:
5496:
5480:
5478:
5477:
5472:
5460:
5458:
5457:
5452:
5450:
5449:
5448:
5428:
5426:
5425:
5420:
5404:
5402:
5401:
5396:
5394:
5393:
5377:
5375:
5374:
5369:
5357:
5355:
5354:
5349:
5337:
5335:
5334:
5329:
5312:partial isometry
5309:
5307:
5306:
5301:
5299:
5298:
5297:
5278:
5276:
5275:
5270:
5265:
5264:
5259:
5244:
5243:
5238:
5228:
5216:
5215:
5199:
5197:
5196:
5191:
5189:
5188:
5187:
5171:
5170:
5154:
5152:
5151:
5146:
5144:
5143:
5136:
5135:
5130:
5117:
5116:
5111:
5084:
5082:
5081:
5076:
5074:
5073:
5068:
5053:
5052:
5047:
5034:
5032:
5031:
5026:
5008:
5006:
5005:
5000:
4988:
4986:
4985:
4980:
4962:
4960:
4959:
4954:
4943:of the subspace
4938:
4936:
4935:
4930:
4928:
4927:
4922:
4907:
4906:
4901:
4877:
4875:
4874:
4869:
4867:
4866:
4861:
4852:
4844:
4836:
4832:
4831:
4827:
4826:
4821:
4811:
4807:
4806:
4805:
4800:
4794:
4793:
4792:
4786:
4764:
4756:
4755:
4750:
4744:
4743:
4742:
4736:
4730:
4722:
4721:
4716:
4710:
4709:
4708:
4702:
4696:
4688:
4683:
4682:
4681:
4664:
4662:
4661:
4656:
4654:
4653:
4648:
4639:
4638:
4633:
4624:
4612:
4610:
4609:
4604:
4602:
4586:
4584:
4583:
4578:
4576:
4560:
4558:
4557:
4552:
4550:
4538:
4536:
4535:
4530:
4525:
4524:
4523:
4517:
4511:
4503:
4502:
4501:
4477:
4475:
4474:
4469:
4467:
4447:
4445:
4444:
4439:
4408:
4406:
4405:
4400:
4398:
4394:
4382:
4378:
4377:
4357:
4355:
4354:
4349:
4347:
4343:
4331:
4327:
4326:
4307:
4299:
4282:
4271:
4257:
4256:
4251:
4247:
4246:
4221:
4219:
4218:
4213:
4211:
4198:bounded operator
4180:
4178:
4177:
4172:
4170:
4169:
4147:
4145:
4144:
4139:
4127:
4125:
4124:
4119:
4107:
4105:
4104:
4099:
4087:
4085:
4084:
4079:
4074:
4069:
4068:
4056:
4042:
4034:
4017:
4006:
3989:
3978:
3963:
3961:
3960:
3955:
3941:
3930:
3919:
3902:
3888:
3877:
3859:
3857:
3856:
3851:
3839:
3837:
3836:
3831:
3813:
3811:
3810:
3805:
3793:
3791:
3790:
3785:
3773:
3771:
3770:
3765:
3741:
3739:
3738:
3733:
3722:
3717:
3713:
3712:
3711:
3688:
3674:
3663:
3655:
3637:
3635:
3634:
3629:
3621:
3610:
3598:
3596:
3595:
3590:
3582:
3567:
3565:
3564:
3559:
3547:
3545:
3544:
3539:
3537:
3525:
3523:
3522:
3517:
3515:
3503:
3501:
3500:
3495:
3477:
3475:
3474:
3469:
3461:
3453:
3436:
3425:
3408:
3397:
3383:. Equivalently:
3382:
3380:
3379:
3374:
3363:
3349:
3338:
3318:
3307:
3296:
3278:
3276:
3275:
3270:
3258:
3256:
3255:
3250:
3248:
3236:
3234:
3233:
3228:
3226:
3210:
3208:
3207:
3202:
3190:
3188:
3187:
3182:
3167:can be used. An
3154:
3152:
3151:
3146:
3088:
3086:
3085:
3080:
3068:
3066:
3065:
3060:
3031:
3030:
3008:
3006:
3005:
3000:
2988:
2986:
2985:
2980:
2960:
2958:
2957:
2952:
2936:
2934:
2933:
2928:
2920:
2918:
2895:
2887:
2879:
2874:
2873:
2839:
2837:
2836:
2831:
2795:
2793:
2792:
2787:
2775:
2773:
2772:
2767:
2755:
2753:
2752:
2747:
2735:
2733:
2732:
2727:
2715:
2713:
2712:
2707:
2695:
2693:
2692:
2687:
2675:
2673:
2672:
2667:
2655:
2653:
2652:
2647:
2635:
2633:
2632:
2627:
2615:
2613:
2612:
2607:
2583:
2581:
2580:
2575:
2556:
2554:
2553:
2548:
2537:
2523:
2511:
2509:
2508:
2503:
2501:
2496:
2492:
2474:
2463:
2455:
2443:
2441:
2440:
2435:
2433:
2422:
2410:
2408:
2407:
2402:
2400:
2392:
2384:
2372:
2370:
2369:
2364:
2356:
2344:
2342:
2341:
2336:
2307:
2305:
2304:
2299:
2294:
2286:
2269:
2254:
2252:
2251:
2246:
2234:
2232:
2231:
2226:
2211:
2209:
2208:
2203:
2188:
2186:
2185:
2180:
2168:
2166:
2165:
2160:
2140:
2138:
2137:
2132:
2120:
2118:
2117:
2112:
2097:
2095:
2094:
2089:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2049:
2029:
2027:
2026:
2021:
2016:
2015:
2014:
1991:
1989:
1988:
1983:
1981:
1966:
1964:
1963:
1958:
1956:
1955:
1954:
1934:
1932:
1931:
1926:
1924:
1912:
1910:
1909:
1904:
1902:
1901:
1900:
1883:
1881:
1880:
1875:
1873:
1833:
1831:
1830:
1825:
1817:
1816:
1796:
1794:
1793:
1788:
1732:
1730:
1729:
1724:
1713:
1712:
1711:
1694:
1692:
1691:
1686:
1665:
1663:
1662:
1657:
1642:
1640:
1639:
1634:
1622:
1620:
1619:
1614:
1603:
1602:
1564:
1563:
1528:
1527:
1489:
1488:
1473:, one sees that
1466:
1464:
1463:
1458:
1453:
1452:
1396:
1394:
1393:
1388:
1380:
1379:
1378:
1359:
1357:
1356:
1351:
1346:
1345:
1307:
1306:
1271:
1270:
1232:
1231:
1199:
1198:
1182:
1180:
1179:
1174:
1172:
1171:
1149:
1147:
1146:
1141:
1127:
1125:
1124:
1119:
1114:
1113:
1078:
1077:
1029:
1027:
1026:
1021:
1016:
1015:
936:
934:
933:
928:
898:
896:
895:
890:
888:
887:
882:
869:
867:
866:
861:
803:
801:
800:
795:
779:
777:
776:
771:
769:
768:
752:
750:
749:
744:
728:
726:
725:
720:
718:
717:
716:
695:
693:
692:
687:
685:
684:
666:
665:
642:
640:
639:
634:
632:
631:
630:
611:
610:
590:
588:
587:
582:
571:A square matrix
567:
565:
564:
559:
551:
550:
530:
528:
527:
522:
494:
492:
491:
486:
478:
470:
458:
456:
455:
450:
445:
434:
420:
412:
395:if it satisfies
390:
388:
387:
382:
370:
368:
367:
362:
342:
340:
339:
334:
314:
312:
311:
306:
291:
289:
288:
283:
275:
274:
258:
256:
255:
250:
226:
224:
223:
218:
178:
176:
175:
170:
158:
156:
155:
150:
138:
136:
135:
130:
98:
96:
95:
90:
21:
14317:
14316:
14312:
14311:
14310:
14308:
14307:
14306:
14282:
14281:
14280:
14275:
14257:
14219:
14175:
14112:
14064:
14006:
13997:
13963:Change of basis
13953:Multilinear map
13891:
13873:
13868:
13818:
13806:
13802:
13792:
13773:. Interscience.
13763:
13747:
13742:
13741:
13708:
13704:
13666:
13662:
13657:
13653:
13647:
13631:
13627:
13597:
13593:
13587:
13571:
13567:
13558:
13554:
13549:
13545:
13539:
13523:
13519:
13514:
13510:
13505:
13501:
13496:
13492:
13485:
13471:
13464:
13459:
13455:
13450:
13411:
13359:
13355:
13341:
13338:
13337:
13306:
13303:
13302:
13299:
13297:Generalizations
13279:measurable sets
13213:
13192:
13189:
13188:
13136:
13133:
13132:
13116:
13113:
13112:
13096:
13093:
13092:
13069:
13065:
13063:
13060:
13059:
13016:
13013:
13012:
13011:. The operator
13001:
12985:
12982:
12981:
12962:
12959:
12958:
12942:
12939:
12938:
12925:, although for
12910:
12907:
12906:
12890:
12887:
12886:
12870:
12867:
12866:
12850:
12847:
12846:
12773:
12770:
12769:
12741:
12738:
12737:
12729:
12715:
12699:
12696:
12695:
12683:
12668:
12661:
12656:
12646:
12630:
12627:
12626:
12614:
12608:
12592:
12589:
12588:
12563:
12560:
12559:
12551:
12546:
12538:
12533:
12513:
12510:
12509:
12493:
12490:
12489:
12473:
12470:
12469:
12455:
12439:
12436:
12435:
12419:
12416:
12415:
12399:
12396:
12395:
12298:
12295:
12294:
12274:
12271:
12270:
12254:
12251:
12250:
12230:
12227:
12226:
12210:
12207:
12206:
12190:
12187:
12186:
12170:
12167:
12166:
12107:
12104:
12103:
12087:
12084:
12083:
12043:
12040:
12039:
12016:
12012:
12010:
12007:
12006:
11984:
11981:
11980:
11945:
11941:
11927:
11924:
11923:
11900:
11896:
11894:
11891:
11890:
11874:
11871:
11870:
11854:
11851:
11850:
11827:
11823:
11821:
11818:
11817:
11801:
11798:
11797:
11781:
11778:
11777:
11740:
11737:
11736:
11708:
11705:
11704:
11688:
11685:
11684:
11668:
11665:
11664:
11641:
11638:
11637:
11617:
11614:
11613:
11610:
11584:
11580:
11578:
11575:
11574:
11552:
11549:
11548:
11524:
11521:
11520:
11503:
11499:
11490:
11486:
11484:
11481:
11480:
11443:
11440:
11439:
11422:
11418:
11416:
11413:
11412:
11384:
11381:
11380:
11354:
11350:
11335:
11331:
11322:
11318:
11316:
11313:
11312:
11289:
11285:
11276:
11272:
11262:
11261:
11256:
11250:
11249:
11243:
11239:
11237:
11227:
11226:
11210:
11209:
11204:
11198:
11197:
11191:
11187:
11185:
11175:
11174:
11166:
11163:
11162:
11127:
11124:
11123:
11101:
11098:
11097:
11070:
11066:
11064:
11061:
11060:
11044:
11041:
11040:
11037:identity matrix
11019:
11015:
11013:
11010:
11009:
10993:
10990:
10989:
10969:
10966:
10965:
10939:
10935:
10926:
10922:
10914:
10911:
10910:
10891:
10888:
10887:
10864:
10860:
10858:
10855:
10854:
10826:
10823:
10822:
10805:
10801:
10793:
10790:
10789:
10788:Any projection
10786:
10784:Canonical forms
10746:
10743:
10742:
10726:
10724:
10721:
10720:
10698:
10696:
10693:
10692:
10670:
10668:
10667:
10665:
10662:
10661:
10645:
10636:
10632:
10630:
10627:
10626:
10610:
10602:
10593:
10589:
10581:
10579:
10576:
10575:
10559:
10557:
10554:
10553:
10532:
10527:
10526:
10517:
10512:
10511:
10502:
10497:
10496:
10495:
10488:
10483:
10482:
10474:
10473:
10471:
10463:
10454:
10450:
10448:
10445:
10444:
10428:
10425:
10424:
10408:
10406:
10403:
10402:
10386:
10383:
10382:
10365:
10360:
10359:
10344:
10339:
10338:
10329:
10324:
10323:
10321:
10318:
10317:
10301:
10298:
10297:
10294:
10255:
10251:
10245:
10241:
10239:
10225:
10221:
10219:
10196:
10193:
10192:
10162:
10159:
10158:
10142:
10139:
10138:
10121:
10120:
10115:
10113:
10107:
10106:
10077:
10071:
10070:
10053:
10045:
10040:
10028:
10021:
10020:
10011:
10007:
10005:
10002:
10001:
9979:
9976:
9975:
9958:
9957:
9952:
9950:
9944:
9943:
9926:
9918:
9913:
9901:
9894:
9893:
9884:
9880:
9878:
9875:
9874:
9858:
9855:
9854:
9837:
9833:
9818:
9814:
9805:
9801:
9799:
9796:
9795:
9778:
9773:
9763:
9759:
9750:
9746:
9737:
9733:
9721:
9716:
9710:
9707:
9706:
9689:
9685:
9683:
9680:
9679:
9658:
9653:
9647:
9644:
9643:
9627:
9624:
9623:
9606:
9602:
9600:
9597:
9596:
9580:
9577:
9576:
9550:
9547:
9546:
9530:
9527:
9526:
9504:
9501:
9500:
9498:
9496:Singular values
9476:
9472:
9461:
9458:
9457:
9441:
9438:
9437:
9420:
9416:
9407:
9403:
9394:
9390:
9378:
9374:
9372:
9369:
9368:
9352:
9349:
9348:
9327:
9323:
9314:
9300:
9296:
9295:
9291:
9290:
9279:
9276:
9275:
9258:
9254:
9252:
9249:
9248:
9227:
9226:
9222:
9214:
9211:
9210:
9192:
9191:
9187:
9178:
9163:
9162:
9158:
9157:
9153:
9152:
9141:
9138:
9137:
9115:
9112:
9111:
9095:
9092:
9091:
9070:
9069:
9065:
9056:
9052:
9042:
9041:
9037:
9023:
9020:
9019:
8997:
8995:
8992:
8991:
8990:for any vector
8975:
8968:
8967:
8963:
8954:
8950:
8940:
8939:
8935:
8921:
8909:
8904:
8903:
8895:
8890:
8887:
8886:
8867:
8860:
8859:
8855:
8846:
8842:
8832:
8831:
8827:
8816:
8814:
8811:
8810:
8794:
8783:
8772:
8762:
8761:
8757:
8755:
8752:
8751:
8730:
8729:
8725:
8723:
8720:
8719:
8697:
8694:
8693:
8677:
8674:
8673:
8657:
8654:
8653:
8637:
8634:
8633:
8617:
8606:
8595:
8585:
8584:
8580:
8578:
8575:
8574:
8558:
8556:
8553:
8552:
8536:
8533:
8532:
8516:
8507:
8502:
8501:
8494:
8493:
8489:
8487:
8484:
8483:
8466:
8461:
8460:
8458:
8455:
8454:
8453:and the vector
8438:
8436:
8433:
8432:
8416:
8413:
8412:
8396:
8384:
8379:
8378:
8376:
8373:
8372:
8353:
8350:
8349:
8332:
8327:
8326:
8324:
8321:
8320:
8301:
8298:
8297:
8281:
8278:
8277:
8260:
8255:
8254:
8252:
8249:
8248:
8232:
8221:
8212:
8208:
8197:
8179:
8174:
8173:
8165:
8162:
8161:
8139:
8125:
8116:
8111:
8110:
8108:
8105:
8104:
8088:
8085:
8084:
8065:
8050:
8045:
8044:
8042:
8039:
8038:
8037:, where vector
8021:
8016:
8015:
8006:
8001:
8000:
7992:
7990:
7987:
7986:
7970:
7967:
7966:
7950:
7948:
7945:
7944:
7921:
7920:
7916:
7907:
7892:
7891:
7887:
7886:
7882:
7881:
7870:
7867:
7866:
7844:
7841:
7840:
7824:
7821:
7820:
7804:
7801:
7800:
7783:
7778:
7777:
7762:
7757:
7756:
7754:
7751:
7750:
7734:
7731:
7730:
7704:
7701:
7700:
7684:
7681:
7680:
7658:
7655:
7654:
7632:
7629:
7628:
7612:
7609:
7608:
7586:
7583:
7582:
7566:
7563:
7562:
7545:
7540:
7539:
7524:
7519:
7518:
7516:
7513:
7512:
7496:
7493:
7492:
7469:
7465:
7463:
7460:
7459:
7431:
7428:
7427:
7424:
7396:
7357:
7356:
7349:
7348:
7344:
7335:
7317:
7316:
7312:
7311:
7307:
7306:
7303:
7302:
7295:
7294:
7290:
7281:
7263:
7262:
7258:
7257:
7253:
7252:
7245:
7244:
7237:
7236:
7231:
7221:
7220:
7212:
7209:
7208:
7192:
7189:
7188:
7161:
7160:
7156:
7149:
7148:
7144:
7128:
7127:
7123:
7121:
7118:
7117:
7101:
7100:
7093:
7092:
7088:
7079:
7064:
7063:
7059:
7058:
7054:
7053:
7040:
7039:
7035:
7026:
7011:
7010:
7006:
7005:
7001:
7000:
6988:
6987:
6980:
6979:
6972:
6971:
6967:
6964:
6963:
6956:
6955:
6951:
6944:
6943:
6934:
6927:
6926:
6916:
6915:
6911:
6909:
6903:
6902:
6897:
6887:
6886:
6882:
6875:
6874:
6873:
6866:
6865:
6860:
6850:
6849:
6840:
6839:
6832:
6831:
6824:
6823:
6819:
6816:
6815:
6808:
6807:
6803:
6796:
6795:
6786:
6774:
6773:
6768:
6758:
6757:
6750:
6749:
6742:
6741:
6737:
6734:
6733:
6726:
6725:
6721:
6714:
6713:
6712:
6708:
6707:
6700:
6699:
6694:
6684:
6683:
6674:
6673:
6666:
6665:
6658:
6657:
6653:
6650:
6649:
6642:
6641:
6637:
6630:
6629:
6620:
6613:
6612:
6605:
6604:
6600:
6597:
6596:
6589:
6588:
6584:
6577:
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6566:
6559:
6558:
6553:
6543:
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6541:
6534:
6533:
6528:
6518:
6517:
6510:
6503:
6501:
6498:
6497:
6481:
6478:
6477:
6457:
6454:
6453:
6426:
6425:
6421:
6419:
6416:
6415:
6397:
6396:
6391:
6381:
6380:
6378:
6375:
6374:
6366:stands for the
6350:
6346:
6344:
6341:
6340:
6323:
6319:
6307:
6303:
6301:
6298:
6297:
6267:
6266:
6262:
6253:
6235:
6234:
6230:
6229:
6225:
6224:
6212:
6208:
6206:
6203:
6202:
6185:
6180:
6165:
6161:
6133:
6129:
6117:
6113:
6111:
6108:
6107:
6090:
6086:
6084:
6081:
6080:
6057:
6053:
6044:
6040:
6026:
6023:
6022:
6006:
6003:
6002:
5979:
5976:
5975:
5957:
5956:
5951:
5950:
5941:
5931:
5924:
5923:
5918:
5917:
5916:
5912:
5911:
5906:
5904:
5901:
5900:
5880:
5871:
5867:
5866:
5858:
5851:
5850:
5845:
5844:
5842:
5839:
5838:
5812:
5808:
5806:
5803:
5802:
5784:
5783:
5778:
5777:
5772:
5770:
5767:
5766:
5742:
5727:
5726:
5722:
5721:
5717:
5716:
5714:
5711:
5710:
5694:
5691:
5690:
5674:
5671:
5670:
5647:
5646:
5642:
5633:
5618:
5617:
5613:
5612:
5608:
5607:
5595:
5591:
5589:
5586:
5585:
5569:
5566:
5565:
5543:
5540:
5539:
5518:
5513:
5512:
5497:
5492:
5491:
5489:
5486:
5485:
5466:
5463:
5462:
5444:
5443:
5439:
5434:
5431:
5430:
5414:
5411:
5410:
5389:
5385:
5383:
5380:
5379:
5363:
5360:
5359:
5343:
5340:
5339:
5323:
5320:
5319:
5293:
5292:
5288:
5286:
5283:
5282:
5260:
5255:
5254:
5239:
5234:
5233:
5224:
5211:
5207:
5205:
5202:
5201:
5183:
5182:
5178:
5166:
5162:
5160:
5157:
5156:
5138:
5137:
5131:
5126:
5125:
5123:
5118:
5112:
5107:
5106:
5099:
5098:
5090:
5087:
5086:
5069:
5064:
5063:
5048:
5043:
5042:
5040:
5037:
5036:
5014:
5011:
5010:
4994:
4991:
4990:
4968:
4965:
4964:
4948:
4945:
4944:
4923:
4918:
4917:
4902:
4897:
4896:
4894:
4891:
4890:
4862:
4857:
4856:
4848:
4840:
4822:
4817:
4816:
4812:
4801:
4796:
4795:
4788:
4787:
4782:
4781:
4780:
4776:
4769:
4765:
4760:
4751:
4746:
4745:
4738:
4737:
4732:
4731:
4726:
4717:
4712:
4711:
4704:
4703:
4698:
4697:
4692:
4684:
4677:
4676:
4672:
4670:
4667:
4666:
4649:
4644:
4643:
4634:
4629:
4628:
4620:
4618:
4615:
4614:
4598:
4596:
4593:
4592:
4572:
4570:
4567:
4566:
4546:
4544:
4541:
4540:
4519:
4518:
4513:
4512:
4507:
4497:
4496:
4492:
4490:
4487:
4486:
4463:
4461:
4458:
4457:
4454:
4421:
4418:
4417:
4390:
4386:
4373:
4369:
4365:
4363:
4360:
4359:
4339:
4335:
4322:
4318:
4314:
4303:
4295:
4278:
4267:
4252:
4242:
4238:
4234:
4233:
4231:
4228:
4227:
4207:
4205:
4202:
4201:
4194:
4165:
4161:
4153:
4150:
4149:
4133:
4130:
4129:
4113:
4110:
4109:
4093:
4090:
4089:
4070:
4064:
4060:
4052:
4038:
4030:
4013:
4002:
3985:
3974:
3969:
3966:
3965:
3937:
3926:
3915:
3898:
3884:
3873:
3865:
3862:
3861:
3845:
3842:
3841:
3819:
3816:
3815:
3799:
3796:
3795:
3779:
3776:
3775:
3747:
3744:
3743:
3718:
3707:
3703:
3696:
3692:
3684:
3670:
3659:
3651:
3643:
3640:
3639:
3617:
3606:
3604:
3601:
3600:
3578:
3573:
3570:
3569:
3553:
3550:
3549:
3533:
3531:
3528:
3527:
3511:
3509:
3506:
3505:
3489:
3486:
3485:
3457:
3449:
3432:
3421:
3404:
3393:
3388:
3385:
3384:
3359:
3345:
3334:
3314:
3303:
3292:
3284:
3281:
3280:
3264:
3261:
3260:
3244:
3242:
3239:
3238:
3222:
3220:
3217:
3216:
3196:
3193:
3192:
3191:and the kernel
3176:
3173:
3172:
3140:
3137:
3136:
3133:
3125:Main articles:
3123:
3099:
3074:
3071:
3070:
3026:
3022:
3020:
3017:
3016:
2994:
2991:
2990:
2974:
2971:
2970:
2946:
2943:
2942:
2899:
2894:
2878:
2866:
2862:
2845:
2842:
2841:
2813:
2810:
2809:
2802:
2781:
2778:
2777:
2761:
2758:
2757:
2741:
2738:
2737:
2721:
2718:
2717:
2701:
2698:
2697:
2681:
2678:
2677:
2661:
2658:
2657:
2641:
2638:
2637:
2621:
2618:
2617:
2616:. The operator
2589:
2586:
2585:
2569:
2566:
2565:
2533:
2519:
2517:
2514:
2513:
2497:
2482:
2478:
2470:
2459:
2451:
2449:
2446:
2445:
2429:
2418:
2416:
2413:
2412:
2396:
2388:
2380:
2378:
2375:
2374:
2352:
2350:
2347:
2346:
2345:. Every vector
2318:
2315:
2314:
2290:
2282:
2265:
2260:
2257:
2256:
2240:
2237:
2236:
2220:
2217:
2216:
2197:
2194:
2193:
2174:
2171:
2170:
2154:
2151:
2150:
2126:
2123:
2122:
2106:
2103:
2102:
2083:
2080:
2079:
2063:
2060:
2059:
2043:
2040:
2039:
2036:
2010:
2009:
2005:
1997:
1994:
1993:
1977:
1972:
1969:
1968:
1950:
1946:
1942:
1940:
1937:
1936:
1920:
1918:
1915:
1914:
1896:
1895:
1891:
1889:
1886:
1885:
1869:
1867:
1864:
1863:
1840:
1812:
1808:
1806:
1803:
1802:
1782:
1779:
1778:
1775:
1756:. The range of
1738:
1707:
1706:
1702:
1700:
1697:
1696:
1674:
1671:
1670:
1651:
1648:
1647:
1646:The projection
1628:
1625:
1624:
1597:
1596:
1591:
1585:
1584:
1579:
1569:
1568:
1558:
1557:
1552:
1546:
1545:
1540:
1530:
1529:
1522:
1521:
1516:
1510:
1509:
1504:
1494:
1493:
1484:
1480:
1478:
1475:
1474:
1447:
1446:
1441:
1435:
1434:
1429:
1419:
1418:
1410:
1407:
1406:
1403:
1374:
1373:
1369:
1367:
1364:
1363:
1362:Observing that
1340:
1339:
1333:
1332:
1326:
1325:
1315:
1314:
1301:
1300:
1294:
1293:
1287:
1286:
1276:
1275:
1265:
1264:
1258:
1257:
1251:
1250:
1240:
1239:
1226:
1225:
1219:
1218:
1212:
1211:
1201:
1200:
1194:
1190:
1188:
1185:
1184:
1167:
1163:
1155:
1152:
1151:
1135:
1132:
1131:
1108:
1107:
1101:
1100:
1094:
1093:
1083:
1082:
1072:
1071:
1065:
1064:
1058:
1057:
1047:
1046:
1041:
1038:
1037:
1010:
1009:
1004:
999:
993:
992:
987:
982:
976:
975:
970:
965:
955:
954:
946:
943:
942:
904:
901:
900:
883:
878:
877:
875:
872:
871:
837:
834:
833:
830:
825:
789:
786:
785:
764:
760:
758:
755:
754:
738:
735:
734:
712:
711:
707:
705:
702:
701:
680:
676:
661:
657:
655:
652:
651:
626:
625:
621:
606:
602:
600:
597:
596:
576:
573:
572:
546:
542:
540:
537:
536:
516:
513:
512:
505:
474:
466:
464:
461:
460:
441:
430:
416:
408:
400:
397:
396:
376:
373:
372:
356:
353:
352:
328:
325:
324:
300:
297:
296:
270:
266:
264:
261:
260:
232:
229:
228:
212:
209:
208:
201:
195:in the object.
164:
161:
160:
144:
141:
140:
112:
109:
108:
84:
81:
80:
39:
28:
23:
22:
15:
12:
11:
5:
14315:
14305:
14304:
14299:
14297:Linear algebra
14294:
14277:
14276:
14274:
14273:
14262:
14259:
14258:
14256:
14255:
14250:
14245:
14240:
14235:
14233:Floating-point
14229:
14227:
14221:
14220:
14218:
14217:
14215:Tensor product
14212:
14207:
14202:
14200:Function space
14197:
14192:
14186:
14184:
14177:
14176:
14174:
14173:
14168:
14163:
14158:
14153:
14148:
14143:
14138:
14136:Triple product
14133:
14128:
14122:
14120:
14114:
14113:
14111:
14110:
14105:
14100:
14095:
14090:
14085:
14080:
14074:
14072:
14066:
14065:
14063:
14062:
14057:
14052:
14050:Transformation
14047:
14042:
14040:Multiplication
14037:
14032:
14027:
14022:
14016:
14014:
14008:
14007:
14000:
13998:
13996:
13995:
13990:
13985:
13980:
13975:
13970:
13965:
13960:
13955:
13950:
13945:
13940:
13935:
13930:
13925:
13920:
13915:
13910:
13905:
13899:
13897:
13896:Basic concepts
13893:
13892:
13890:
13889:
13884:
13878:
13875:
13874:
13871:Linear algebra
13867:
13866:
13859:
13852:
13844:
13838:
13837:
13831:
13828:Pavel Grinfeld
13815:
13801:
13800:External links
13798:
13797:
13796:
13790:
13775:
13766:
13762:978-1420095388
13761:
13746:
13743:
13740:
13739:
13720:(1): 220–224.
13702:
13676:(2): 852–870,
13660:
13651:
13646:978-1420095388
13645:
13625:
13607:(5): 375–381,
13591:
13586:978-1420095388
13585:
13565:
13552:
13543:
13538:978-1420095388
13537:
13517:
13508:
13499:
13490:
13483:
13462:
13452:
13451:
13449:
13446:
13445:
13444:
13439:
13434:
13429:
13424:
13418:
13410:
13407:
13370:
13367:
13362:
13358:
13354:
13351:
13348:
13345:
13325:
13322:
13319:
13316:
13313:
13310:
13298:
13295:
13275:measure theory
13263:
13262:
13256:
13251:
13240:
13235:
13212:
13209:
13196:
13176:
13173:
13170:
13167:
13164:
13161:
13158:
13155:
13152:
13149:
13146:
13143:
13140:
13131:and therefore
13120:
13100:
13080:
13077:
13072:
13068:
13047:
13044:
13041:
13038:
13035:
13032:
13029:
13026:
13023:
13020:
12989:
12966:
12946:
12927:Hilbert spaces
12914:
12894:
12874:
12854:
12831:
12828:
12825:
12822:
12819:
12816:
12813:
12810:
12807:
12804:
12801:
12798:
12795:
12792:
12789:
12786:
12783:
12780:
12777:
12757:
12754:
12751:
12748:
12745:
12727:
12714:is closed and
12703:
12681:
12666:
12659:
12634:
12612:
12607:is closed and
12596:
12576:
12573:
12570:
12567:
12549:
12536:
12517:
12497:
12477:
12443:
12423:
12403:
12380:
12377:
12374:
12371:
12368:
12365:
12362:
12359:
12356:
12353:
12350:
12347:
12344:
12341:
12338:
12335:
12332:
12329:
12326:
12323:
12320:
12317:
12314:
12311:
12308:
12305:
12302:
12278:
12258:
12234:
12214:
12194:
12174:
12147:
12144:
12141:
12138:
12135:
12132:
12129:
12126:
12123:
12120:
12117:
12114:
12111:
12091:
12071:
12068:
12065:
12062:
12059:
12056:
12053:
12050:
12047:
12027:
12024:
12019:
12015:
11994:
11991:
11988:
11968:
11965:
11962:
11959:
11956:
11953:
11948:
11944:
11940:
11937:
11934:
11931:
11911:
11908:
11903:
11899:
11878:
11858:
11838:
11835:
11830:
11826:
11805:
11785:
11765:
11762:
11759:
11756:
11753:
11750:
11747:
11744:
11724:
11721:
11718:
11715:
11712:
11692:
11672:
11645:
11621:
11609:
11606:
11587:
11583:
11562:
11559:
11556:
11528:
11506:
11502:
11498:
11493:
11489:
11468:
11465:
11462:
11459:
11456:
11453:
11450:
11447:
11425:
11421:
11400:
11397:
11394:
11391:
11388:
11365:
11362:
11357:
11353:
11349:
11346:
11343:
11338:
11334:
11330:
11325:
11321:
11309:
11308:
11297:
11292:
11288:
11284:
11279:
11275:
11271:
11266:
11260:
11257:
11255:
11252:
11251:
11246:
11242:
11238:
11236:
11233:
11232:
11230:
11225:
11222:
11219:
11214:
11208:
11205:
11203:
11200:
11199:
11194:
11190:
11186:
11184:
11181:
11180:
11178:
11173:
11170:
11131:
11111:
11108:
11105:
11079:
11076:
11073:
11069:
11048:
11022:
11018:
10997:
10973:
10962:
10961:
10948:
10945:
10942:
10938:
10934:
10929:
10925:
10921:
10918:
10895:
10875:
10872:
10867:
10863:
10830:
10808:
10804:
10800:
10797:
10785:
10782:
10750:
10729:
10701:
10677:
10673:
10648:
10644:
10639:
10635:
10613:
10609:
10605:
10601:
10596:
10592:
10588:
10584:
10562:
10552:). The vector
10535:
10530:
10520:
10515:
10510:
10505:
10500:
10491:
10486:
10481:
10477:
10470:
10466:
10462:
10457:
10453:
10432:
10411:
10390:
10368:
10363:
10358:
10355:
10352:
10347:
10342:
10337:
10332:
10327:
10305:
10293:
10290:
10277:
10274:
10271:
10268:
10265:
10258:
10254:
10248:
10244:
10238:
10233:
10228:
10224:
10218:
10215:
10212:
10209:
10206:
10203:
10200:
10172:
10169:
10166:
10146:
10124:
10114:
10112:
10109:
10108:
10105:
10102:
10099:
10096:
10093:
10090:
10087:
10084:
10081:
10078:
10076:
10073:
10072:
10069:
10066:
10063:
10060:
10057:
10054:
10048:
10043:
10039:
10035:
10032:
10027:
10026:
10024:
10019:
10014:
10010:
9989:
9986:
9983:
9961:
9951:
9949:
9946:
9945:
9942:
9939:
9936:
9933:
9930:
9927:
9921:
9916:
9912:
9908:
9905:
9900:
9899:
9897:
9892:
9887:
9883:
9862:
9840:
9836:
9832:
9829:
9826:
9821:
9817:
9813:
9808:
9804:
9781:
9776:
9772:
9766:
9762:
9756:
9753:
9749:
9745:
9740:
9736:
9732:
9729:
9724:
9719:
9715:
9692:
9688:
9661:
9656:
9652:
9631:
9609:
9605:
9584:
9560:
9557:
9554:
9534:
9514:
9511:
9508:
9497:
9494:
9479:
9475:
9471:
9468:
9465:
9445:
9423:
9419:
9413:
9410:
9406:
9402:
9397:
9393:
9389:
9386:
9381:
9377:
9356:
9347:of the matrix
9330:
9326:
9320:
9317:
9312:
9308:
9303:
9299:
9294:
9289:
9286:
9283:
9261:
9257:
9230:
9225:
9221:
9218:
9195:
9190:
9184:
9181:
9176:
9172:
9166:
9161:
9156:
9151:
9148:
9145:
9125:
9122:
9119:
9099:
9073:
9068:
9062:
9059:
9055:
9051:
9045:
9040:
9036:
9033:
9030:
9027:
9007:
9004:
9000:
8978:
8971:
8966:
8960:
8957:
8953:
8949:
8943:
8938:
8934:
8931:
8928:
8924:
8920:
8917:
8912:
8907:
8902:
8898:
8894:
8874:
8870:
8863:
8858:
8852:
8849:
8845:
8841:
8835:
8830:
8826:
8823:
8819:
8797:
8793:
8790:
8786:
8782:
8779:
8775:
8771:
8765:
8760:
8739:
8733:
8728:
8707:
8704:
8701:
8681:
8661:
8641:
8620:
8616:
8613:
8609:
8605:
8602:
8598:
8594:
8588:
8583:
8561:
8540:
8519:
8515:
8510:
8505:
8497:
8492:
8469:
8464:
8441:
8420:
8399:
8395:
8392:
8387:
8382:
8360:
8357:
8335:
8330:
8308:
8305:
8285:
8263:
8258:
8235:
8231:
8228:
8224:
8220:
8215:
8211:
8207:
8204:
8200:
8196:
8193:
8190:
8187:
8182:
8177:
8172:
8169:
8149:
8146:
8142:
8138:
8135:
8132:
8128:
8124:
8119:
8114:
8092:
8072:
8068:
8064:
8061:
8058:
8053:
8048:
8024:
8019:
8014:
8009:
8004:
7999:
7995:
7974:
7953:
7930:
7924:
7919:
7913:
7910:
7905:
7901:
7895:
7890:
7885:
7880:
7877:
7874:
7865:) is given by
7854:
7851:
7848:
7828:
7808:
7786:
7781:
7776:
7773:
7770:
7765:
7760:
7738:
7714:
7711:
7708:
7688:
7668:
7665:
7662:
7642:
7639:
7636:
7616:
7596:
7593:
7590:
7570:
7548:
7543:
7538:
7535:
7532:
7527:
7522:
7500:
7480:
7477:
7472:
7468:
7447:
7444:
7441:
7438:
7435:
7423:
7420:
7395:
7392:
7369:
7366:
7361:
7352:
7347:
7341:
7338:
7333:
7329:
7326:
7320:
7315:
7310:
7305:
7304:
7298:
7293:
7287:
7284:
7279:
7275:
7272:
7266:
7261:
7256:
7251:
7250:
7248:
7241:
7235:
7232:
7230:
7227:
7226:
7224:
7219:
7216:
7196:
7176:
7173:
7170:
7164:
7159:
7152:
7147:
7143:
7140:
7137:
7131:
7126:
7096:
7091:
7085:
7082:
7077:
7073:
7067:
7062:
7057:
7052:
7049:
7043:
7038:
7032:
7029:
7024:
7020:
7014:
7009:
7004:
6999:
6996:
6993:
6991:
6989:
6984:
6975:
6970:
6966:
6965:
6959:
6954:
6950:
6949:
6947:
6940:
6937:
6931:
6925:
6919:
6914:
6910:
6908:
6905:
6904:
6901:
6898:
6896:
6890:
6885:
6881:
6880:
6878:
6870:
6864:
6861:
6859:
6856:
6855:
6853:
6848:
6845:
6843:
6841:
6836:
6827:
6822:
6818:
6817:
6811:
6806:
6802:
6801:
6799:
6792:
6789:
6784:
6778:
6772:
6769:
6767:
6764:
6763:
6761:
6754:
6745:
6740:
6736:
6735:
6729:
6724:
6720:
6719:
6717:
6711:
6704:
6698:
6695:
6693:
6690:
6689:
6687:
6682:
6679:
6677:
6675:
6670:
6661:
6656:
6652:
6651:
6645:
6640:
6636:
6635:
6633:
6626:
6623:
6617:
6608:
6603:
6599:
6598:
6592:
6587:
6583:
6582:
6580:
6572:
6569:
6563:
6557:
6554:
6552:
6549:
6548:
6546:
6538:
6532:
6529:
6527:
6524:
6523:
6521:
6516:
6513:
6511:
6509:
6506:
6505:
6485:
6461:
6441:
6438:
6435:
6429:
6424:
6401:
6395:
6392:
6390:
6387:
6386:
6384:
6353:
6349:
6326:
6322:
6318:
6315:
6310:
6306:
6279:
6276:
6270:
6265:
6259:
6256:
6251:
6247:
6244:
6238:
6233:
6228:
6223:
6220:
6215:
6211:
6188:
6183:
6178:
6174:
6171:
6168:
6164:
6159:
6154:
6151:
6148:
6145:
6142:
6139:
6136:
6132:
6128:
6125:
6120:
6116:
6093:
6089:
6068:
6065:
6060:
6056:
6052:
6047:
6043:
6039:
6036:
6033:
6030:
6010:
5983:
5960:
5954:
5947:
5944:
5939:
5934:
5927:
5921:
5915:
5909:
5888:
5883:
5878:
5874:
5870:
5865:
5861:
5854:
5848:
5826:
5823:
5819:
5815:
5811:
5787:
5781:
5775:
5748:
5745:
5740:
5736:
5730:
5725:
5720:
5698:
5678:
5656:
5650:
5645:
5639:
5636:
5631:
5627:
5621:
5616:
5611:
5606:
5603:
5598:
5594:
5573:
5553:
5550:
5547:
5521:
5516:
5511:
5508:
5505:
5500:
5495:
5470:
5447:
5442:
5438:
5418:
5392:
5388:
5367:
5347:
5327:
5296:
5291:
5268:
5263:
5258:
5253:
5250:
5247:
5242:
5237:
5232:
5227:
5223:
5219:
5214:
5210:
5186:
5181:
5177:
5174:
5169:
5165:
5142:
5134:
5129:
5124:
5122:
5119:
5115:
5110:
5105:
5104:
5102:
5097:
5094:
5072:
5067:
5062:
5059:
5056:
5051:
5046:
5024:
5021:
5018:
4998:
4978:
4975:
4972:
4952:
4926:
4921:
4916:
4913:
4910:
4905:
4900:
4865:
4860:
4855:
4851:
4847:
4843:
4839:
4835:
4830:
4825:
4820:
4815:
4810:
4804:
4799:
4791:
4785:
4779:
4775:
4772:
4768:
4763:
4759:
4754:
4749:
4741:
4735:
4729:
4725:
4720:
4715:
4707:
4701:
4695:
4691:
4687:
4680:
4675:
4652:
4647:
4642:
4637:
4632:
4627:
4623:
4601:
4575:
4549:
4528:
4522:
4516:
4510:
4506:
4500:
4495:
4466:
4453:
4450:
4437:
4434:
4431:
4428:
4425:
4397:
4393:
4389:
4385:
4381:
4376:
4372:
4368:
4346:
4342:
4338:
4334:
4330:
4325:
4321:
4317:
4313:
4310:
4306:
4302:
4298:
4294:
4291:
4288:
4285:
4281:
4277:
4274:
4270:
4266:
4263:
4260:
4255:
4250:
4245:
4241:
4237:
4210:
4193:
4190:
4168:
4164:
4160:
4157:
4137:
4117:
4097:
4077:
4073:
4067:
4063:
4059:
4055:
4051:
4048:
4045:
4041:
4037:
4033:
4029:
4026:
4023:
4020:
4016:
4012:
4009:
4005:
4001:
3998:
3995:
3992:
3988:
3984:
3981:
3977:
3973:
3953:
3950:
3947:
3944:
3940:
3936:
3933:
3929:
3925:
3922:
3918:
3914:
3911:
3908:
3905:
3901:
3897:
3894:
3891:
3887:
3883:
3880:
3876:
3872:
3869:
3849:
3829:
3826:
3823:
3803:
3783:
3763:
3760:
3757:
3754:
3751:
3731:
3728:
3725:
3721:
3716:
3710:
3706:
3702:
3699:
3695:
3691:
3687:
3683:
3680:
3677:
3673:
3669:
3666:
3662:
3658:
3654:
3650:
3647:
3627:
3624:
3620:
3616:
3613:
3609:
3588:
3585:
3581:
3577:
3557:
3536:
3514:
3493:
3467:
3464:
3460:
3456:
3452:
3448:
3445:
3442:
3439:
3435:
3431:
3428:
3424:
3420:
3417:
3414:
3411:
3407:
3403:
3400:
3396:
3392:
3372:
3369:
3366:
3362:
3358:
3355:
3352:
3348:
3344:
3341:
3337:
3333:
3330:
3327:
3324:
3321:
3317:
3313:
3310:
3306:
3302:
3299:
3295:
3291:
3288:
3268:
3247:
3225:
3200:
3180:
3144:
3122:
3119:
3098:
3095:
3091:diagonalizable
3078:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3029:
3025:
2998:
2978:
2950:
2926:
2923:
2917:
2914:
2911:
2908:
2905:
2902:
2898:
2893:
2890:
2885:
2882:
2877:
2872:
2869:
2865:
2861:
2858:
2855:
2852:
2849:
2829:
2826:
2823:
2820:
2817:
2801:
2798:
2785:
2765:
2745:
2725:
2705:
2685:
2665:
2645:
2625:
2605:
2602:
2599:
2596:
2593:
2573:
2558:
2557:
2546:
2543:
2540:
2536:
2532:
2529:
2526:
2522:
2500:
2495:
2491:
2488:
2485:
2481:
2477:
2473:
2469:
2466:
2462:
2458:
2454:
2432:
2428:
2425:
2421:
2399:
2395:
2391:
2387:
2383:
2362:
2359:
2355:
2334:
2331:
2328:
2325:
2322:
2308:
2297:
2293:
2289:
2285:
2281:
2278:
2275:
2272:
2268:
2264:
2244:
2224:
2201:
2178:
2158:
2130:
2110:
2098:. Suppose the
2087:
2067:
2047:
2035:
2032:
2019:
2013:
2008:
2004:
2001:
1980:
1976:
1953:
1949:
1945:
1923:
1899:
1894:
1872:
1839:
1836:
1823:
1820:
1815:
1811:
1786:
1774:
1771:
1737:
1734:
1722:
1719:
1716:
1710:
1705:
1684:
1681:
1678:
1668:if and only if
1666:is orthogonal
1655:
1632:
1612:
1609:
1606:
1601:
1595:
1592:
1590:
1587:
1586:
1583:
1580:
1578:
1575:
1574:
1572:
1567:
1562:
1556:
1553:
1551:
1548:
1547:
1544:
1541:
1539:
1536:
1535:
1533:
1526:
1520:
1517:
1515:
1512:
1511:
1508:
1505:
1503:
1500:
1499:
1497:
1492:
1487:
1483:
1456:
1451:
1445:
1442:
1440:
1437:
1436:
1433:
1430:
1428:
1425:
1424:
1422:
1417:
1414:
1402:
1399:
1386:
1383:
1377:
1372:
1349:
1344:
1338:
1335:
1334:
1331:
1328:
1327:
1324:
1321:
1320:
1318:
1313:
1310:
1305:
1299:
1296:
1295:
1292:
1289:
1288:
1285:
1282:
1281:
1279:
1274:
1269:
1263:
1260:
1259:
1256:
1253:
1252:
1249:
1246:
1245:
1243:
1238:
1235:
1230:
1224:
1221:
1220:
1217:
1214:
1213:
1210:
1207:
1206:
1204:
1197:
1193:
1170:
1166:
1162:
1159:
1139:
1117:
1112:
1106:
1103:
1102:
1099:
1096:
1095:
1092:
1089:
1088:
1086:
1081:
1076:
1070:
1067:
1066:
1063:
1060:
1059:
1056:
1053:
1052:
1050:
1045:
1019:
1014:
1008:
1005:
1003:
1000:
998:
995:
994:
991:
988:
986:
983:
981:
978:
977:
974:
971:
969:
966:
964:
961:
960:
958:
953:
950:
926:
923:
920:
917:
914:
911:
908:
886:
881:
859:
856:
853:
850:
847:
844:
841:
829:
826:
824:
821:
813:
812:
805:
793:
767:
763:
742:
715:
710:
700:matrix, where
683:
679:
675:
672:
669:
664:
660:
629:
624:
620:
617:
614:
609:
605:
580:
569:
557:
554:
549:
545:
520:
504:
501:
484:
481:
477:
473:
469:
448:
444:
440:
437:
433:
429:
426:
423:
419:
415:
411:
407:
404:
380:
360:
332:
304:
281:
278:
273:
269:
248:
245:
242:
239:
236:
216:
200:
197:
168:
148:
128:
125:
122:
119:
116:
103:to itself (an
88:
66:linear algebra
26:
9:
6:
4:
3:
2:
14314:
14303:
14300:
14298:
14295:
14293:
14290:
14289:
14287:
14272:
14264:
14263:
14260:
14254:
14251:
14249:
14248:Sparse matrix
14246:
14244:
14241:
14239:
14236:
14234:
14231:
14230:
14228:
14226:
14222:
14216:
14213:
14211:
14208:
14206:
14203:
14201:
14198:
14196:
14193:
14191:
14188:
14187:
14185:
14183:constructions
14182:
14178:
14172:
14171:Outermorphism
14169:
14167:
14164:
14162:
14159:
14157:
14154:
14152:
14149:
14147:
14144:
14142:
14139:
14137:
14134:
14132:
14131:Cross product
14129:
14127:
14124:
14123:
14121:
14119:
14115:
14109:
14106:
14104:
14101:
14099:
14098:Outer product
14096:
14094:
14091:
14089:
14086:
14084:
14081:
14079:
14078:Orthogonality
14076:
14075:
14073:
14071:
14067:
14061:
14058:
14056:
14055:Cramer's rule
14053:
14051:
14048:
14046:
14043:
14041:
14038:
14036:
14033:
14031:
14028:
14026:
14025:Decomposition
14023:
14021:
14018:
14017:
14015:
14013:
14009:
14004:
13994:
13991:
13989:
13986:
13984:
13981:
13979:
13976:
13974:
13971:
13969:
13966:
13964:
13961:
13959:
13956:
13954:
13951:
13949:
13946:
13944:
13941:
13939:
13936:
13934:
13931:
13929:
13926:
13924:
13921:
13919:
13916:
13914:
13911:
13909:
13906:
13904:
13901:
13900:
13898:
13894:
13888:
13885:
13883:
13880:
13879:
13876:
13872:
13865:
13860:
13858:
13853:
13851:
13846:
13845:
13842:
13835:
13832:
13829:
13825:
13821:
13816:
13813:
13809:
13804:
13803:
13793:
13787:
13783:
13782:
13776:
13772:
13767:
13764:
13758:
13754:
13749:
13748:
13735:
13731:
13727:
13723:
13719:
13715:
13714:
13706:
13699:
13695:
13691:
13687:
13683:
13679:
13675:
13671:
13664:
13655:
13648:
13642:
13638:
13637:
13629:
13622:
13618:
13614:
13610:
13606:
13602:
13595:
13588:
13582:
13578:
13577:
13569:
13562:
13556:
13547:
13540:
13534:
13530:
13529:
13521:
13512:
13506:Meyer, p. 431
13503:
13497:Meyer, p. 433
13494:
13486:
13484:9780521839402
13480:
13476:
13469:
13467:
13457:
13453:
13443:
13440:
13438:
13435:
13433:
13430:
13428:
13425:
13422:
13419:
13416:
13413:
13412:
13406:
13404:
13400:
13396:
13392:
13388:
13384:
13368:
13360:
13352:
13349:
13346:
13323:
13320:
13314:
13311:
13308:
13294:
13292:
13288:
13284:
13280:
13276:
13272:
13268:
13261:
13257:
13255:
13252:
13249:
13245:
13242:Reduction to
13241:
13239:
13236:
13233:
13229:
13225:
13222:
13221:
13220:
13218:
13208:
13194:
13171:
13168:
13165:
13159:
13156:
13153:
13147:
13141:
13138:
13118:
13098:
13078:
13075:
13070:
13066:
13045:
13039:
13033:
13030:
13024:
13018:
13008:
13004:
12987:
12980:
12964:
12944:
12936:
12932:
12928:
12912:
12892:
12872:
12852:
12843:
12829:
12826:
12823:
12820:
12817:
12814:
12811:
12808:
12805:
12802:
12799:
12796:
12793:
12787:
12784:
12781:
12775:
12755:
12752:
12749:
12746:
12743:
12734:
12730:
12723:
12719:
12701:
12692:
12688:
12684:
12677:
12673:
12669:
12662:
12653:
12649:
12632:
12624:
12619:
12615:
12594:
12574:
12571:
12568:
12565:
12556:
12552:
12543:
12539:
12531:
12515:
12495:
12475:
12466:
12462:
12458:
12441:
12421:
12401:
12392:
12375:
12369:
12366:
12363:
12357:
12354:
12351:
12345:
12342:
12339:
12333:
12327:
12324:
12321:
12315:
12309:
12306:
12303:
12300:
12292:
12276:
12256:
12248:
12232:
12212:
12192:
12172:
12164:
12159:
12142:
12139:
12136:
12130:
12127:
12124:
12118:
12112:
12109:
12089:
12066:
12063:
12060:
12054:
12051:
12048:
12045:
12025:
12022:
12017:
12013:
11992:
11989:
11986:
11963:
11960:
11957:
11951:
11946:
11938:
11935:
11932:
11909:
11906:
11901:
11897:
11876:
11856:
11836:
11833:
11828:
11824:
11803:
11783:
11763:
11760:
11754:
11751:
11748:
11742:
11722:
11719:
11716:
11713:
11710:
11690:
11670:
11661:
11659:
11643:
11635:
11619:
11605:
11603:
11585:
11581:
11560:
11557:
11554:
11546:
11542:
11526:
11504:
11500:
11496:
11491:
11487:
11479:. The factor
11466:
11463:
11460:
11457:
11454:
11451:
11448:
11445:
11423:
11419:
11398:
11395:
11392:
11389:
11386:
11379:
11363:
11360:
11355:
11351:
11347:
11344:
11341:
11336:
11332:
11328:
11323:
11319:
11295:
11290:
11286:
11282:
11277:
11273:
11269:
11264:
11258:
11253:
11244:
11240:
11234:
11228:
11223:
11220:
11217:
11212:
11206:
11201:
11192:
11188:
11182:
11176:
11171:
11168:
11161:
11160:
11159:
11157:
11153:
11149:
11148:inner product
11145:
11129:
11109:
11106:
11103:
11095:
11077:
11074:
11071:
11067:
11046:
11038:
11020:
11016:
10995:
10987:
10971:
10946:
10943:
10940:
10936:
10932:
10927:
10923:
10919:
10916:
10909:
10908:
10907:
10906:has the form
10893:
10873:
10870:
10865:
10861:
10852:
10848:
10844:
10828:
10806:
10802:
10798:
10795:
10778:
10774:
10770:
10766:
10764:
10748:
10718:
10717:
10642:
10637:
10633:
10607:
10599:
10594:
10590:
10586:
10551:
10533:
10518:
10508:
10503:
10489:
10479:
10468:
10460:
10455:
10451:
10430:
10388:
10366:
10356:
10353:
10350:
10345:
10335:
10330:
10303:
10289:
10272:
10266:
10263:
10256:
10252:
10246:
10242:
10236:
10231:
10226:
10222:
10216:
10210:
10207:
10204:
10198:
10190:
10186:
10170:
10167:
10164:
10144:
10110:
10103:
10100:
10097:
10094:
10091:
10088:
10085:
10082:
10079:
10074:
10067:
10064:
10061:
10058:
10055:
10046:
10041:
10037:
10033:
10030:
10022:
10017:
10012:
10008:
9987:
9984:
9981:
9947:
9940:
9937:
9934:
9931:
9928:
9919:
9914:
9910:
9906:
9903:
9895:
9890:
9885:
9881:
9860:
9838:
9834:
9830:
9827:
9824:
9819:
9815:
9811:
9806:
9802:
9779:
9774:
9770:
9764:
9760:
9754:
9751:
9743:
9738:
9734:
9727:
9722:
9717:
9713:
9690:
9686:
9677:
9659:
9654:
9650:
9629:
9607:
9603:
9582:
9574:
9558:
9555:
9552:
9532:
9512:
9509:
9506:
9493:
9477:
9473:
9469:
9466:
9463:
9443:
9421:
9417:
9411:
9408:
9400:
9395:
9391:
9384:
9379:
9375:
9354:
9346:
9328:
9324:
9318:
9315:
9310:
9306:
9301:
9297:
9292:
9287:
9284:
9281:
9259:
9255:
9247:
9223:
9219:
9216:
9188:
9182:
9179:
9174:
9170:
9159:
9154:
9149:
9146:
9143:
9123:
9120:
9117:
9097:
9088:
9066:
9060:
9057:
9049:
9038:
9031:
9028:
9025:
9005:
9002:
8964:
8958:
8955:
8947:
8936:
8929:
8926:
8918:
8915:
8910:
8900:
8892:
8885:In this way,
8872:
8856:
8850:
8847:
8839:
8828:
8821:
8791:
8780:
8777:
8758:
8737:
8726:
8705:
8702:
8699:
8679:
8659:
8639:
8614:
8603:
8600:
8581:
8538:
8513:
8508:
8490:
8467:
8418:
8393:
8390:
8385:
8358:
8355:
8333:
8306:
8303:
8283:
8261:
8229:
8213:
8209:
8205:
8191:
8188:
8180:
8167:
8147:
8133:
8130:
8122:
8117:
8103:, and vector
8090:
8059:
8056:
8051:
8022:
8012:
8007:
7997:
7972:
7941:
7928:
7917:
7911:
7908:
7903:
7899:
7888:
7883:
7878:
7875:
7872:
7852:
7849:
7846:
7826:
7806:
7784:
7774:
7771:
7768:
7763:
7736:
7728:
7712:
7709:
7706:
7686:
7666:
7663:
7660:
7640:
7637:
7634:
7614:
7594:
7591:
7588:
7568:
7546:
7536:
7533:
7530:
7525:
7498:
7478:
7475:
7470:
7466:
7445:
7439:
7436:
7433:
7419:
7415:
7413:
7409:
7405:
7401:
7391:
7389:
7385:
7380:
7367:
7364:
7359:
7345:
7339:
7336:
7331:
7327:
7324:
7313:
7308:
7291:
7285:
7282:
7277:
7273:
7270:
7259:
7254:
7246:
7239:
7233:
7228:
7222:
7217:
7214:
7194:
7174:
7171:
7168:
7157:
7145:
7141:
7138:
7135:
7124:
7114:
7089:
7083:
7080:
7075:
7071:
7060:
7055:
7050:
7047:
7036:
7030:
7027:
7022:
7018:
7007:
7002:
6997:
6994:
6992:
6982:
6968:
6952:
6945:
6938:
6935:
6929:
6923:
6912:
6906:
6899:
6894:
6883:
6876:
6868:
6862:
6857:
6851:
6846:
6844:
6834:
6820:
6804:
6797:
6790:
6787:
6782:
6776:
6770:
6765:
6759:
6752:
6738:
6722:
6715:
6709:
6702:
6696:
6691:
6685:
6680:
6678:
6668:
6654:
6638:
6631:
6624:
6621:
6615:
6601:
6585:
6578:
6570:
6567:
6561:
6555:
6550:
6544:
6536:
6530:
6525:
6519:
6514:
6512:
6507:
6483:
6475:
6459:
6439:
6436:
6433:
6422:
6399:
6393:
6388:
6382:
6371:
6369:
6351:
6347:
6324:
6320:
6316:
6313:
6308:
6304:
6295:
6290:
6277:
6274:
6263:
6257:
6254:
6249:
6245:
6242:
6231:
6226:
6221:
6218:
6213:
6209:
6186:
6181:
6172:
6169:
6166:
6157:
6149:
6143:
6140:
6137:
6134:
6130:
6126:
6123:
6118:
6114:
6091:
6087:
6066:
6063:
6058:
6054:
6050:
6045:
6037:
6034:
6031:
6008:
6000:
5995:
5981:
5945:
5942:
5937:
5913:
5886:
5881:
5863:
5824:
5821:
5764:
5746:
5743:
5738:
5734:
5723:
5718:
5696:
5689:still embeds
5676:
5667:
5654:
5643:
5637:
5634:
5629:
5625:
5614:
5609:
5604:
5601:
5596:
5592:
5571:
5551:
5548:
5545:
5537:
5519:
5509:
5506:
5503:
5498:
5482:
5468:
5440:
5436:
5416:
5408:
5390:
5386:
5365:
5345:
5325:
5317:
5313:
5289:
5279:
5266:
5261:
5248:
5245:
5240:
5225:
5221:
5217:
5212:
5208:
5179:
5175:
5172:
5167:
5163:
5140:
5132:
5120:
5113:
5100:
5095:
5092:
5070:
5060:
5057:
5054:
5049:
5022:
5019:
5016:
4996:
4976:
4973:
4970:
4950:
4942:
4924:
4914:
4911:
4908:
4903:
4888:
4883:
4881:
4863:
4853:
4845:
4837:
4833:
4823:
4808:
4802:
4777:
4773:
4770:
4766:
4757:
4752:
4723:
4718:
4689:
4673:
4650:
4640:
4635:
4625:
4590:
4564:
4526:
4504:
4493:
4485:
4484:outer product
4481:
4449:
4432:
4429:
4426:
4415:
4410:
4383:
4370:
4332:
4319:
4311:
4300:
4292:
4286:
4275:
4272:
4264:
4258:
4253:
4239:
4225:
4199:
4189:
4187:
4182:
4166:
4162:
4158:
4155:
4135:
4115:
4095:
4065:
4061:
4057:
4046:
4035:
4027:
4021:
4010:
4007:
3999:
3993:
3982:
3979:
3951:
3948:
3934:
3931:
3920:
3912:
3906:
3895:
3892:
3881:
3878:
3847:
3827:
3824:
3821:
3801:
3794:. Therefore,
3781:
3758:
3755:
3752:
3729:
3726:
3714:
3708:
3704:
3700:
3697:
3693:
3689:
3678:
3667:
3664:
3656:
3648:
3625:
3622:
3614:
3611:
3586:
3583:
3575:
3555:
3491:
3483:
3478:
3465:
3454:
3446:
3440:
3429:
3426:
3418:
3412:
3401:
3398:
3370:
3367:
3356:
3353:
3342:
3339:
3325:
3311:
3308:
3297:
3289:
3266:
3214:
3198:
3178:
3170:
3166:
3165:orthogonality
3162:
3161:Hilbert space
3158:
3157:inner product
3142:
3132:
3128:
3118:
3116:
3115:endomorphisms
3110:
3108:
3104:
3094:
3092:
3076:
3053:
3050:
3047:
3041:
3038:
3035:
3032:
3027:
3023:
3015:
3010:
2996:
2976:
2968:
2964:
2948:
2940:
2924:
2921:
2912:
2909:
2906:
2900:
2896:
2891:
2888:
2883:
2880:
2875:
2870:
2867:
2859:
2856:
2853:
2850:
2824:
2821:
2818:
2807:
2797:
2783:
2763:
2743:
2723:
2703:
2683:
2663:
2643:
2623:
2603:
2600:
2597:
2594:
2591:
2571:
2563:
2562:complementary
2544:
2541:
2538:
2530:
2527:
2524:
2493:
2489:
2486:
2483:
2479:
2475:
2467:
2464:
2456:
2426:
2423:
2393:
2385:
2360:
2357:
2332:
2329:
2326:
2323:
2320:
2313:
2309:
2295:
2287:
2279:
2276:
2273:
2270:
2242:
2222:
2215:
2199:
2192:
2191:
2190:
2176:
2156:
2148:
2144:
2128:
2108:
2101:
2085:
2065:
2045:
2031:
2006:
1999:
1974:
1947:
1943:
1892:
1884:and any ball
1861:
1857:
1853:
1849:
1845:
1835:
1821:
1818:
1813:
1809:
1800:
1784:
1767:
1763:
1759:
1755:
1751:
1747:
1742:
1733:
1720:
1717:
1714:
1703:
1682:
1679:
1676:
1669:
1653:
1644:
1630:
1623:showing that
1610:
1607:
1604:
1599:
1593:
1588:
1581:
1576:
1570:
1565:
1560:
1554:
1549:
1542:
1537:
1531:
1524:
1518:
1513:
1506:
1501:
1495:
1490:
1485:
1481:
1472:
1467:
1454:
1449:
1443:
1438:
1431:
1426:
1420:
1415:
1412:
1398:
1384:
1381:
1370:
1360:
1347:
1342:
1336:
1329:
1322:
1316:
1311:
1308:
1303:
1297:
1290:
1283:
1277:
1272:
1267:
1261:
1254:
1247:
1241:
1236:
1233:
1228:
1222:
1215:
1208:
1202:
1195:
1191:
1183:, we compute
1168:
1164:
1160:
1157:
1137:
1128:
1115:
1110:
1104:
1097:
1090:
1084:
1079:
1074:
1068:
1061:
1054:
1048:
1043:
1035:
1030:
1017:
1012:
1006:
1001:
996:
989:
984:
979:
972:
967:
962:
956:
951:
948:
940:
921:
918:
915:
912:
909:
899:to the point
884:
854:
851:
848:
845:
842:
820:
818:
810:
806:
791:
783:
765:
761:
740:
732:
708:
699:
681:
677:
673:
670:
667:
662:
658:
649:
646:
622:
618:
615:
612:
607:
603:
594:
591:is called an
578:
570:
555:
552:
547:
543:
534:
518:
511:
510:square matrix
507:
506:
500:
498:
482:
479:
471:
438:
435:
424:
413:
405:
394:
391:is called an
378:
358:
350:
349:orthogonality
346:
345:Hilbert space
330:
322:
318:
317:inner product
302:
293:
279:
276:
271:
267:
246:
240:
237:
234:
214:
206:
196:
194:
190:
186:
182:
166:
146:
126:
123:
120:
117:
114:
106:
102:
86:
79:
75:
71:
67:
59:
56:
52:
48:
43:
37:
33:
19:
14181:Vector space
13937:
13913:Vector space
13780:
13770:
13752:
13717:
13711:
13705:
13673:
13669:
13663:
13654:
13635:
13628:
13604:
13600:
13594:
13575:
13568:
13555:
13546:
13527:
13520:
13511:
13502:
13493:
13474:
13456:
13394:
13390:
13300:
13264:
13214:
13006:
13002:
12844:
12732:
12725:
12721:
12717:
12690:
12686:
12679:
12675:
12671:
12664:
12657:
12651:
12647:
12622:
12617:
12610:
12554:
12547:
12541:
12534:
12464:
12460:
12456:
12393:
12290:
12246:
12160:
11662:
11658:Banach space
11611:
11604:components.
11601:
11544:
11540:
11310:
11155:
11151:
10963:
10849:, since its
10787:
10776:
10772:
10714:
10295:
9499:
9089:
7942:
7653:, otherwise
7425:
7416:
7399:
7397:
7381:
7115:
6372:
6291:
6106:is given by
5996:
5765:-1 operator
5668:
5483:
5406:
5280:
4884:
4588:
4562:
4455:
4411:
4195:
4183:
3482:self-adjoint
3479:
3168:
3134:
3111:
3100:
3011:
2961:is always a
2803:
2561:
2559:
2512:, and where
2037:
1841:
1776:
1765:
1761:
1757:
1753:
1749:
1745:
1645:
1468:
1404:
1361:
1130:To see that
1129:
1031:
938:
831:
814:
808:
729:denotes the
592:
532:
531:is called a
496:
392:
323:, i.e. when
294:
204:
202:
107:) such that
105:endomorphism
101:vector space
73:
63:
57:
46:
14161:Multivector
14126:Determinant
14083:Dot product
13928:Linear span
12508:and kernel
12488:with range
12293:subspaces:
12249:projection
11796:and kernel
11539:acts as an
11152:orthonormal
11094:zero matrix
10185:matrix norm
8247:, and then
5669:The matrix
5407:final space
5281:The matrix
5009:denote the
4880:dot product
4480:unit vector
2967:eigenspaces
1773:Idempotence
817:eigenvalues
199:Definitions
14286:Categories
14195:Direct sum
14030:Invertible
13933:Linear map
13745:References
13244:Hessenberg
13217:algorithms
13058:satisfies
13000:such that
12736:, we have
12694:. Because
12532:. Suppose
12454:such that
12247:continuous
12163:continuous
11573:) and the
11541:orthogonal
11144:direct sum
9018:and hence
8482:satisfies
6476:matrix of
6474:null space
4989:, and let
4088:for every
3504:, for any
2939:eigenvalue
2312:direct sum
2310:We have a
1799:idempotent
259:such that
205:projection
181:idempotent
74:projection
14225:Numerical
13988:Transpose
13734:122704926
13698:219921214
13621:122277398
13559:See also
13366:→
13361:⊥
13350:
13318:→
13312::
13169:−
13160:
13142:
13099:φ
13034:φ
12988:φ
12821:−
12803:−
12785:−
12753:∈
12747:−
12370:
12364:⊕
12355:−
12346:
12328:
12322:⊕
12310:
12140:−
12131:
12125:⊕
12113:
12064:−
11990:−
11961:−
11936:−
11720:⊕
11582:σ
11497:⊕
11420:σ
11352:σ
11348:≥
11345:⋯
11342:≥
11333:σ
11329:≥
11320:σ
11283:⊕
11270:⊕
11241:σ
11224:⊕
11221:⋯
11218:⊕
11189:σ
11130:⊕
11107:−
11075:−
10944:−
10933:⊕
10871:−
10676:^
10643:
10600:
10509:⋅
10480:⋅
10461:
10354:…
10267:κ
10253:σ
10243:σ
10237:≥
10223:σ
10208:−
10199:κ
10168:−
10117:otherwise
10101:−
10095:≤
10089:≤
10065:≤
10059:≤
10038:γ
10009:σ
9985:−
9954:otherwise
9938:≤
9932:≤
9911:γ
9882:σ
9835:γ
9831:≥
9828:…
9825:≥
9816:γ
9812:≥
9803:γ
9780:⊥
9752:−
9660:⊥
9556:−
9510:−
9422:∗
9409:−
9396:∗
9329:∗
9316:−
9302:∗
9260:∗
9180:−
9058:−
9003:∈
8956:−
8848:−
8778:−
8703:×
8601:−
8573:so that
8411:for some
8206:−
8131:−
7909:−
7850:≥
7772:…
7710:−
7638:≥
7592:×
7534:…
7443:→
7437::
7398:The term
7337:−
7283:−
7081:−
7028:−
6936:−
6788:−
6622:−
6568:−
6255:−
6170:−
6158:
6144:
6138:∈
6059:†
6042:⟩
6029:⟨
5943:−
5822:≠
5744:−
5635:−
5549:≥
5507:…
5252:⟩
5249:⋅
5231:⟨
5222:∑
5121:⋯
5058:…
5020:×
4974:≥
4912:…
4887:dimension
4864:∥
4846:⋅
4824:∥
4803:∥
4774:
4753:⊥
4719:∥
4651:⊥
4636:∥
4436:⟩
4433:⋅
4427:⋅
4424:⟨
4384:≤
4333:⋅
4312:≤
4309:⟩
4290:⟨
4284:⟩
4262:⟨
4167:∗
4076:⟩
4066:∗
4050:⟨
4044:⟩
4025:⟨
4019:⟩
3997:⟨
3991:⟩
3972:⟨
3946:⟩
3932:−
3910:⟨
3904:⟩
3879:−
3868:⟨
3825:−
3762:⟩
3759:⋅
3753:⋅
3750:⟨
3724:⟩
3701:−
3682:⟨
3676:⟩
3665:−
3646:⟨
3623:∈
3612:−
3584:∈
3463:⟩
3444:⟨
3438:⟩
3416:⟨
3410:⟩
3391:⟨
3365:⟩
3340:−
3329:⟨
3323:⟩
3309:−
3287:⟨
3051:−
3033:−
2910:−
2907:λ
2901:λ
2884:λ
2868:−
2857:−
2851:λ
2601:−
2564:, as are
2539:∈
2525:∈
2487:−
2465:−
2358:∈
2330:⊕
2271:∈
2263:∀
2100:subspaces
1677:α
1589:α
1550:α
1514:α
1439:α
766:∗
731:transpose
682:∗
480:∈
447:⟩
428:⟨
422:⟩
403:⟨
244:→
238::
118:∘
14271:Category
14210:Subspace
14205:Quotient
14156:Bivector
14070:Bilinear
14012:Matrices
13887:Glossary
13409:See also
13273:, while
12655:. Also,
12625:lies in
12587:. Since
12038:implies
11378:integers
11096:of size
11039:of size
11008:. Here
10853:divides
9642:and let
8718:-matrix
6177:‖
6163:‖
5877:‖
5869:‖
5818:‖
5810:‖
5085:, i.e.,
4829:‖
4814:‖
4452:Formulas
4396:‖
4388:‖
4380:‖
4367:‖
4345:‖
4337:‖
4329:‖
4316:‖
4249:‖
4236:‖
3568:we have
3107:converse
2806:spectrum
2800:Spectrum
2141:are the
1848:open set
1844:open map
1838:Open map
823:Examples
459:for all
321:complete
13882:Outline
13824:YouTube
13812:YouTube
13690:1680061
13291:lattice
12768:, i.e.
12645:, i.e.
11889:, i.e.
11602:oblique
11142:is the
11092:is the
11035:is the
10984:is the
10841:over a
10719:) from
9674:be the
9595:. Let
7627:. Then
7607:matrix
6472:is the
6452:(i.e.,
6339:. Here
6201:. Then
6001:matrix
5310:is the
4148:; thus
3155:has an
3103:commute
2212:is the
1858:of the
1850:in the
698:complex
319:and is
315:has an
99:from a
14166:Tensor
13978:Kernel
13908:Vector
13903:Scalar
13788:
13759:
13732:
13696:
13688:
13643:
13619:
13583:
13535:
13481:
12937:. Let
12291:closed
11376:. The
11311:where
11122:, and
10964:where
10381:. Let
9873:are:
9436:since
7749:. Let
6131:argmin
5564:, and
5338:, and
4939:be an
4889:. Let
3742:where
3638:, and
2147:kernel
1852:domain
1801:(i.e.
1034:vector
696:for a
648:matrix
643:for a
193:points
14035:Minor
14020:Block
13958:Basis
13826:, by
13730:S2CID
13694:S2CID
13617:S2CID
13448:Notes
13226:(see
13009:) = 1
11656:is a
10845:is a
10843:field
10423:onto
7187:with
6294:frame
6141:range
5538:with
5536:basis
4478:is a
4358:Thus
2776:onto
2716:onto
2411:with
2143:image
1860:image
1752:onto
343:is a
295:When
185:image
76:is a
14190:Dual
14045:Rank
13786:ISBN
13757:ISBN
13686:OSTI
13641:ISBN
13581:ISBN
13533:ISBN
13479:ISBN
13387:onto
13230:and
12865:and
12731:} ⊂
12616:} ⊂
12545:and
12082:and
11361:>
10986:rank
10634:proj
10591:proj
10452:proj
10296:Let
10157:and
10000:are
9545:and
8652:and
7679:and
7426:Let
5763:rank
4539:(If
4108:and
3814:and
3526:and
3237:and
3211:are
3129:and
2584:and
2444:and
2145:and
2121:and
2038:Let
1469:Via
815:The
753:and
645:real
72:, a
68:and
55:line
53:the
51:onto
13822:on
13810:on
13722:doi
13678:doi
13609:doi
13397:In
13347:ker
13139:ker
12670:= (
12367:ker
12343:ker
12325:ker
12185:of
11158:is
10988:of
10741:to
10443:as
9678:of
9575:of
9367:by
8371:so
8160:So
6373:If
5409:of
5318:of
4771:sgn
4128:in
3964:to
3548:in
3259:in
3089:is
2840:as
2235:on
2149:of
1834:).
1797:is
1760:is
1036:is
784:of
733:of
595:if
179:is
64:In
14288::
13728:.
13718:42
13716:.
13692:,
13684:,
13674:41
13672:,
13615:,
13605:35
13603:,
13465:^
13405:.
13395:V.
13234:);
13207:.
13157:rg
12720:−
12716:{(
12689:−
12685:→
12674:−
12665:Px
12663:−
12650:=
12648:Py
12621:,
12611:Px
12553:→
12548:Px
12540:→
12463:⊕
12459:=
12391:.
12307:rg
12158:.
12128:rg
12110:rg
11660:.
11059:,
10765:.
10625:.
9492:.
9087:.
7390:.
5994:.
5825:1.
5481:.
4448:.
4409:.
4226::
4188:.
4181:.
3599:,
3279:,
3093:.
3009:.
2796:.
2255::
2030:.
939:xy
508:A
499:.
292:.
203:A
13863:e
13856:t
13849:v
13830:.
13794:.
13736:.
13724::
13680::
13611::
13563:.
13487:.
13391:W
13369:W
13357:)
13353:T
13344:(
13324:,
13321:W
13315:V
13309:T
13250:)
13195:U
13175:)
13172:P
13166:I
13163:(
13154:=
13151:)
13148:P
13145:(
13119:P
13079:P
13076:=
13071:2
13067:P
13046:u
13043:)
13040:x
13037:(
13031:=
13028:)
13025:x
13022:(
13019:P
13007:u
13005:(
13003:φ
12965:u
12945:U
12913:V
12893:U
12873:V
12853:U
12830:0
12827:=
12824:y
12818:x
12815:P
12812:=
12809:y
12806:P
12800:x
12797:P
12794:=
12791:)
12788:y
12782:x
12779:(
12776:P
12756:V
12750:y
12744:x
12733:V
12728:n
12726:x
12724:)
12722:P
12718:I
12702:V
12691:y
12687:x
12682:n
12680:x
12678:)
12676:P
12672:I
12667:n
12660:n
12658:x
12652:y
12633:U
12623:y
12618:U
12613:n
12609:{
12595:U
12575:y
12572:=
12569:x
12566:P
12555:y
12550:n
12542:x
12537:n
12535:x
12516:V
12496:U
12476:P
12465:V
12461:U
12457:X
12442:V
12422:X
12402:U
12379:)
12376:P
12373:(
12361:)
12358:P
12352:1
12349:(
12340:=
12337:)
12334:P
12331:(
12319:)
12316:P
12313:(
12304:=
12301:X
12277:X
12257:P
12233:P
12213:U
12193:X
12173:U
12146:)
12143:P
12137:1
12134:(
12122:)
12119:P
12116:(
12090:X
12070:)
12067:P
12061:1
12058:(
12055:+
12052:P
12049:=
12046:1
12026:P
12023:=
12018:2
12014:P
11993:P
11987:1
11967:)
11964:P
11958:1
11955:(
11952:=
11947:2
11943:)
11939:P
11933:1
11930:(
11910:P
11907:=
11902:2
11898:P
11877:X
11857:P
11837:P
11834:=
11829:2
11825:P
11804:V
11784:U
11764:u
11761:=
11758:)
11755:v
11752:+
11749:u
11746:(
11743:P
11723:V
11717:U
11714:=
11711:X
11691:X
11671:X
11644:X
11620:X
11586:i
11561:0
11558:=
11555:k
11545:P
11527:P
11505:s
11501:0
11492:m
11488:I
11467:d
11464:=
11461:m
11458:+
11455:s
11452:+
11449:k
11446:2
11424:i
11399:m
11396:,
11393:s
11390:,
11387:k
11364:0
11356:k
11337:2
11324:1
11296:.
11291:s
11287:0
11278:m
11274:I
11265:]
11259:0
11254:0
11245:k
11235:1
11229:[
11213:]
11207:0
11202:0
11193:1
11183:1
11177:[
11172:=
11169:P
11156:P
11110:r
11104:d
11078:r
11072:d
11068:0
11047:r
11021:r
11017:I
10996:P
10972:r
10947:r
10941:d
10937:0
10928:r
10924:I
10920:=
10917:P
10894:P
10874:x
10866:2
10862:x
10829:d
10807:2
10803:P
10799:=
10796:P
10779:.
10777:V
10773:y
10749:V
10728:y
10700:z
10672:y
10647:y
10638:V
10612:z
10608:+
10604:y
10595:V
10587:=
10583:y
10561:y
10534:i
10529:u
10519:i
10514:u
10504:i
10499:u
10490:i
10485:u
10476:y
10469:=
10465:y
10456:V
10431:V
10410:y
10389:y
10367:p
10362:u
10357:,
10351:,
10346:2
10341:u
10336:,
10331:1
10326:u
10304:V
10276:)
10273:P
10270:(
10264:=
10257:k
10247:1
10232:1
10227:1
10217:=
10214:)
10211:P
10205:I
10202:(
10171:P
10165:I
10145:P
10111:0
10104:k
10098:n
10092:i
10086:1
10083:+
10080:k
10075:1
10068:k
10062:i
10056:1
10047:2
10042:i
10034:+
10031:1
10023:{
10018:=
10013:i
9988:P
9982:I
9948:0
9941:k
9935:i
9929:1
9920:2
9915:i
9907:+
9904:1
9896:{
9891:=
9886:i
9861:P
9839:k
9820:2
9807:1
9775:A
9771:Q
9765:T
9761:B
9755:1
9748:)
9744:A
9739:T
9735:B
9731:(
9728:A
9723:T
9718:A
9714:Q
9691:A
9687:Q
9655:A
9651:Q
9630:A
9608:A
9604:Q
9583:A
9559:P
9553:I
9533:P
9513:P
9507:I
9478:+
9474:A
9470:A
9467:=
9464:P
9444:A
9418:A
9412:1
9405:)
9401:A
9392:A
9388:(
9385:=
9380:+
9376:A
9355:A
9325:A
9319:1
9311:)
9307:A
9298:A
9293:(
9288:A
9285:=
9282:P
9256:A
9229:T
9224:P
9220:=
9217:P
9194:T
9189:A
9183:1
9175:)
9171:A
9165:T
9160:A
9155:(
9150:A
9147:=
9144:P
9124:B
9121:=
9118:A
9098:P
9072:T
9067:B
9061:1
9054:)
9050:A
9044:T
9039:B
9035:(
9032:A
9029:=
9026:P
9006:V
8999:x
8977:x
8970:T
8965:B
8959:1
8952:)
8948:A
8942:T
8937:B
8933:(
8930:A
8927:=
8923:w
8919:A
8916:=
8911:1
8906:x
8901:=
8897:x
8893:P
8873:.
8869:x
8862:T
8857:B
8851:1
8844:)
8840:A
8834:T
8829:B
8825:(
8822:=
8818:w
8796:0
8792:=
8789:)
8785:w
8781:A
8774:x
8770:(
8764:T
8759:B
8738:A
8732:T
8727:B
8706:k
8700:k
8680:k
8660:B
8640:A
8619:0
8615:=
8612:)
8608:w
8604:A
8597:x
8593:(
8587:T
8582:B
8560:w
8539:B
8518:0
8514:=
8509:2
8504:x
8496:T
8491:B
8468:2
8463:x
8440:w
8419:k
8398:w
8394:A
8391:=
8386:1
8381:x
8359:,
8356:A
8334:1
8329:x
8307:.
8304:A
8284:P
8262:2
8257:x
8234:0
8230:=
8227:)
8223:x
8219:(
8214:2
8210:P
8203:)
8199:x
8195:(
8192:P
8189:=
8186:)
8181:2
8176:x
8171:(
8168:P
8148:.
8145:)
8141:x
8137:(
8134:P
8127:x
8123:=
8118:2
8113:x
8091:P
8071:)
8067:x
8063:(
8060:P
8057:=
8052:1
8047:x
8023:2
8018:x
8013:+
8008:1
8003:x
7998:=
7994:x
7973:V
7952:x
7929:.
7923:T
7918:B
7912:1
7904:)
7900:A
7894:T
7889:B
7884:(
7879:A
7876:=
7873:P
7853:1
7847:k
7827:P
7807:B
7785:k
7780:v
7775:,
7769:,
7764:1
7759:v
7737:k
7713:k
7707:n
7687:P
7667:0
7664:=
7661:k
7641:1
7635:k
7615:A
7595:k
7589:n
7569:P
7547:k
7542:u
7537:,
7531:,
7526:1
7521:u
7499:P
7479:P
7476:=
7471:2
7467:P
7446:V
7440:V
7434:P
7368:.
7365:W
7360:]
7351:T
7346:B
7340:1
7332:)
7328:B
7325:W
7319:T
7314:B
7309:(
7297:T
7292:A
7286:1
7278:)
7274:A
7271:W
7265:T
7260:A
7255:(
7247:[
7240:]
7234:B
7229:A
7223:[
7218:=
7215:I
7195:W
7175:0
7172:=
7169:B
7163:T
7158:W
7151:T
7146:A
7142:=
7139:B
7136:W
7130:T
7125:A
7095:T
7090:B
7084:1
7076:)
7072:B
7066:T
7061:B
7056:(
7051:B
7048:+
7042:T
7037:A
7031:1
7023:)
7019:A
7013:T
7008:A
7003:(
6998:A
6995:=
6983:]
6974:T
6969:B
6958:T
6953:A
6946:[
6939:1
6930:]
6924:B
6918:T
6913:B
6907:O
6900:O
6895:A
6889:T
6884:A
6877:[
6869:]
6863:B
6858:A
6852:[
6847:=
6835:]
6826:T
6821:B
6810:T
6805:A
6798:[
6791:1
6783:)
6777:]
6771:B
6766:A
6760:[
6753:]
6744:T
6739:B
6728:T
6723:A
6716:[
6710:(
6703:]
6697:B
6692:A
6686:[
6681:=
6669:]
6660:T
6655:B
6644:T
6639:A
6632:[
6625:1
6616:]
6607:T
6602:B
6591:T
6586:A
6579:[
6571:1
6562:]
6556:B
6551:A
6545:[
6537:]
6531:B
6526:A
6520:[
6515:=
6508:I
6484:A
6460:B
6440:0
6437:=
6434:B
6428:T
6423:A
6400:]
6394:B
6389:A
6383:[
6352:+
6348:A
6325:+
6321:A
6317:A
6314:=
6309:A
6305:P
6278:.
6275:D
6269:T
6264:A
6258:1
6250:)
6246:A
6243:D
6237:T
6232:A
6227:(
6222:A
6219:=
6214:A
6210:P
6187:2
6182:D
6173:y
6167:x
6153:)
6150:A
6147:(
6135:y
6127:=
6124:x
6119:A
6115:P
6092:A
6088:P
6067:x
6064:D
6055:y
6051:=
6046:D
6038:y
6035:,
6032:x
6009:D
5982:u
5959:T
5953:u
5946:1
5938:)
5933:u
5926:T
5920:u
5914:(
5908:u
5887:,
5882:2
5873:u
5864:=
5860:u
5853:T
5847:u
5814:u
5786:T
5780:u
5774:u
5747:1
5739:)
5735:A
5729:T
5724:A
5719:(
5697:U
5677:A
5655:.
5649:T
5644:A
5638:1
5630:)
5626:A
5620:T
5615:A
5610:(
5605:A
5602:=
5597:A
5593:P
5572:A
5552:1
5546:k
5520:k
5515:u
5510:,
5504:,
5499:1
5494:u
5469:U
5446:T
5441:A
5437:A
5417:A
5391:A
5387:P
5366:U
5346:A
5326:U
5295:T
5290:A
5267:.
5262:i
5257:u
5246:,
5241:i
5236:u
5226:i
5218:=
5213:A
5209:P
5185:T
5180:A
5176:A
5173:=
5168:A
5164:P
5141:]
5133:k
5128:u
5114:1
5109:u
5101:[
5096:=
5093:A
5071:k
5066:u
5061:,
5055:,
5050:1
5045:u
5023:k
5017:n
4997:A
4977:1
4971:k
4951:U
4925:k
4920:u
4915:,
4909:,
4904:1
4899:u
4859:x
4854:=
4850:0
4842:u
4838:+
4834:)
4819:x
4809:)
4798:x
4790:T
4784:u
4778:(
4767:(
4762:u
4758:=
4748:x
4740:T
4734:u
4728:u
4724:+
4714:x
4706:T
4700:u
4694:u
4690:=
4686:x
4679:u
4674:P
4646:x
4641:+
4631:x
4626:=
4622:x
4600:x
4589:u
4574:u
4563:u
4548:u
4527:.
4521:T
4515:u
4509:u
4505:=
4499:u
4494:P
4465:u
4430:,
4392:v
4375:v
4371:P
4341:v
4324:v
4320:P
4305:v
4301:,
4297:v
4293:P
4287:=
4280:v
4276:P
4273:,
4269:v
4265:P
4259:=
4254:2
4244:v
4240:P
4209:v
4163:P
4159:=
4156:P
4136:W
4116:y
4096:x
4072:y
4062:P
4058:,
4054:x
4047:=
4040:y
4036:,
4032:x
4028:P
4022:=
4015:y
4011:P
4008:,
4004:x
4000:P
3994:=
3987:y
3983:P
3980:,
3976:x
3952:0
3949:=
3943:)
3939:y
3935:P
3928:y
3924:(
3921:,
3917:x
3913:P
3907:=
3900:y
3896:P
3893:,
3890:)
3886:x
3882:P
3875:x
3871:(
3848:P
3828:P
3822:I
3802:P
3782:W
3756:,
3730:0
3727:=
3720:y
3715:)
3709:2
3705:P
3698:P
3694:(
3690:,
3686:x
3679:=
3672:y
3668:P
3661:y
3657:,
3653:x
3649:P
3626:V
3619:y
3615:P
3608:y
3587:U
3580:x
3576:P
3556:W
3535:y
3513:x
3492:P
3466:.
3459:y
3455:,
3451:x
3447:P
3441:=
3434:y
3430:P
3427:,
3423:x
3419:P
3413:=
3406:y
3402:P
3399:,
3395:x
3371:0
3368:=
3361:y
3357:P
3354:,
3351:)
3347:x
3343:P
3336:x
3332:(
3326:=
3320:)
3316:y
3312:P
3305:y
3301:(
3298:,
3294:x
3290:P
3267:W
3246:y
3224:x
3199:V
3179:U
3143:W
3077:P
3057:)
3054:1
3048:x
3045:(
3042:x
3039:=
3036:x
3028:2
3024:x
2997:V
2977:V
2949:P
2925:.
2922:P
2916:)
2913:1
2904:(
2897:1
2892:+
2889:I
2881:1
2876:=
2871:1
2864:)
2860:P
2854:I
2848:(
2828:}
2825:1
2822:,
2819:0
2816:{
2784:V
2764:U
2744:Q
2724:U
2704:V
2684:P
2664:Q
2644:P
2624:Q
2604:P
2598:I
2595:=
2592:Q
2572:P
2545:.
2542:V
2535:v
2531:,
2528:U
2521:u
2499:x
2494:)
2490:P
2484:I
2480:(
2476:=
2472:x
2468:P
2461:x
2457:=
2453:v
2431:x
2427:P
2424:=
2420:u
2398:v
2394:+
2390:u
2386:=
2382:x
2361:W
2354:x
2333:V
2327:U
2324:=
2321:W
2296:.
2292:x
2288:=
2284:x
2280:P
2277::
2274:U
2267:x
2243:U
2223:I
2200:P
2177:P
2157:P
2129:V
2109:U
2086:W
2066:P
2046:W
2018:)
2012:x
2007:B
2003:(
2000:P
1979:x
1975:P
1952:x
1948:P
1944:B
1922:x
1898:x
1893:B
1871:x
1822:P
1819:=
1814:2
1810:P
1785:P
1768:.
1766:k
1762:m
1758:T
1754:m
1750:k
1746:T
1721:.
1718:P
1715:=
1709:T
1704:P
1683:0
1680:=
1654:P
1631:P
1611:.
1608:P
1605:=
1600:]
1594:1
1582:0
1577:0
1571:[
1566:=
1561:]
1555:1
1543:0
1538:0
1532:[
1525:]
1519:1
1507:0
1502:0
1496:[
1491:=
1486:2
1482:P
1455:.
1450:]
1444:1
1432:0
1427:0
1421:[
1416:=
1413:P
1385:P
1382:=
1376:T
1371:P
1348:.
1343:]
1337:z
1330:y
1323:x
1317:[
1312:P
1309:=
1304:]
1298:0
1291:y
1284:x
1278:[
1273:=
1268:]
1262:0
1255:y
1248:x
1242:[
1237:P
1234:=
1229:]
1223:z
1216:y
1209:x
1203:[
1196:2
1192:P
1169:2
1165:P
1161:=
1158:P
1138:P
1116:.
1111:]
1105:0
1098:y
1091:x
1085:[
1080:=
1075:]
1069:z
1062:y
1055:x
1049:[
1044:P
1018:.
1013:]
1007:0
1002:0
997:0
990:0
985:1
980:0
973:0
968:0
963:1
957:[
952:=
949:P
925:)
922:0
919:,
916:y
913:,
910:x
907:(
885:3
880:R
858:)
855:z
852:,
849:y
846:,
843:x
840:(
811:.
804:.
792:P
762:P
741:P
714:T
709:P
678:P
674:=
671:P
668:=
663:2
659:P
628:T
623:P
619:=
616:P
613:=
608:2
604:P
579:P
568:.
556:P
553:=
548:2
544:P
519:P
483:V
476:y
472:,
468:x
443:y
439:P
436:,
432:x
425:=
418:y
414:,
410:x
406:P
379:V
359:P
331:V
303:V
280:P
277:=
272:2
268:P
247:V
241:V
235:P
215:V
167:P
147:P
127:P
124:=
121:P
115:P
87:P
60:.
58:m
47:P
38:.
20:)
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