802:
5041:
1273:
1059:
1268:{\displaystyle {\begin{aligned}&&\mathbf {A} ^{\textsf {T}}\mathbf {b} &-\mathbf {A} ^{\textsf {T}}\mathbf {Ax} =0\\\Rightarrow &&\mathbf {A} ^{\textsf {T}}\mathbf {b} &=\mathbf {A} ^{\textsf {T}}\mathbf {Ax} \\\Rightarrow &&\mathbf {x} &=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}\mathbf {b} \end{aligned}}}
2084:
1834:
2204:
646:
441:
1708:
3038:
1926:
3373:
3631:
1960:
1719:
1465:
790:
2364:
2096:
1522:
3279:
2477:
557:
3534:
2949:
346:
1047:
2423:
1620:
221:
2956:
720:
3433:
2538:
508:
3691:
2736:
2696:
2079:{\displaystyle {\hat {\mathbf {\beta } }}_{\text{GLS}}=\left(\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {y} }
1845:
3284:
1064:
3542:
3114:
2599:
1829:{\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} {\hat {\boldsymbol {\beta }}}=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {y} .}
3797:
The hat matrix was introduced by John Wilder in 1972. An article by
Hoaglin, D.C. and Welsch, R.E. (1978) gives the properties of the matrix and also many examples of its application.
2896:
1395:
253:
173:
672:
2252:
3868:
94:
59:
1302:
963:
3787:
3765:
3739:
3713:
3226:
3177:
3155:
3061:
2860:
2838:
2800:
2764:
2646:
2621:
2562:
2500:
2294:
1547:
1390:
1368:
1346:
1324:
985:
938:
916:
891:
869:
847:
825:
545:
466:
338:
305:
275:
144:
2199:{\displaystyle \mathbf {H} =\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {\Sigma } ^{-1}}
728:
2299:
1481:
3231:
641:{\displaystyle \mathbf {\Sigma } _{\mathbf {r} }=\left(\mathbf {I} -\mathbf {P} \right)^{\textsf {T}}\mathbf {\Sigma } \left(\mathbf {I} -\mathbf {P} \right)}
2428:
3441:
436:{\displaystyle \mathbf {r} =\mathbf {y} -\mathbf {\hat {y}} =\mathbf {y} -\mathbf {P} \mathbf {y} =\left(\mathbf {I} -\mathbf {P} \right)\mathbf {y} .}
4699:
2901:
3816:
3180:
3157:, which is the number of independent parameters of the linear model. For other models such as LOESS that are still linear in the observations
993:
1703:{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}\mathbf {y} ,}
2376:
3715:
is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the
4913:
4132:
3033:{\displaystyle \left(\mathbf {I} -\mathbf {P} \right)\mathbf {P} =\mathbf {P} \left(\mathbf {I} -\mathbf {P} \right)=\mathbf {0} .}
181:
3875:
1946:
The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the
5004:
679:
685:
4098:
3378:
2505:
1921:{\displaystyle \mathbf {P} :=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}.}
475:
3636:
3368:{\displaystyle \mathbf {P} :=\mathbf {X} \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}}
4923:
4689:
3626:{\displaystyle \mathbf {P} =\mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}}
2701:
2661:
2367:
1611:
1573:
316:
4047:
4014:
3989:
3852:
3082:
2567:
4724:
2865:
105:
4271:
987:. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
1460:{\displaystyle \mathbf {A} \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}\mathbf {A} ^{\textsf {T}}}
5077:
4488:
4125:
3745:
of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of
801:
4563:
3821:
3806:
229:
149:
20:
1475:
Suppose that we wish to estimate a linear model using linear least squares. The model can be written as
678:
of the error vector (and by extension, the response vector as well). For the case of linear models with
655:
4719:
4241:
3962:
3896:
2212:
112:, which describe the influence each response value has on the fitted value for that same observation.
4823:
4694:
4608:
3904:
1941:
108:
each response value has on each fitted value. The diagonal elements of the projection matrix are the
71:
36:
4928:
4818:
4526:
4206:
1282:
943:
785:{\displaystyle \mathbf {\Sigma } _{\mathbf {r} }=\left(\mathbf {I} -\mathbf {P} \right)\sigma ^{2}}
3770:
3748:
3722:
3696:
3209:
3160:
3138:
3044:
2843:
2821:
2783:
2747:
2629:
2604:
2545:
2483:
2359:{\displaystyle \left(\mathbf {X} ^{\textsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\textsf {T}}}
2277:
1530:
1373:
1351:
1329:
1307:
968:
921:
899:
874:
852:
830:
808:
528:
449:
321:
288:
258:
127:
4963:
4892:
4774:
4634:
4231:
4118:
3195:
4833:
4416:
4221:
4090:
3128:
1937:
1605:
4039:
3842:
4779:
4516:
4366:
4361:
4196:
4171:
4166:
3811:
3187:
3132:
2267:
1517:{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},}
109:
4082:
4973:
4331:
4161:
4141:
3693:. There are a number of applications of such a decomposition. In the classical application
1550:
8:
4994:
4968:
4546:
4351:
4341:
3716:
3274:{\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}}
1572:
Many types of models and techniques are subject to this formulation. A few examples are
5045:
4999:
4989:
4943:
4938:
4867:
4803:
4669:
4406:
4401:
4336:
4326:
4191:
3978:
3931:
3191:
1581:
2262:
The projection matrix has a number of useful algebraic properties. In the language of
5082:
5056:
5040:
4843:
4838:
4808:
4769:
4764:
4593:
4588:
4573:
4568:
4559:
4554:
4501:
4396:
4346:
4291:
4261:
4256:
4236:
4226:
4186:
4094:
4083:
4043:
4032:
4010:
3985:
3954:
3848:
3076:
2472:{\displaystyle \mathbf {u} =\mathbf {y} -\mathbf {P} \mathbf {y} \perp \mathbf {X} .}
1947:
1589:
1577:
675:
548:
522:
121:
97:
5051:
5019:
4948:
4887:
4882:
4862:
4798:
4704:
4674:
4659:
4639:
4578:
4531:
4506:
4496:
4467:
4386:
4381:
4356:
4286:
4266:
4176:
4156:
3921:
3913:
3869:"Data Assimilation: Observation influence diagnostic of a data assimilation system"
3117:
3072:
1585:
4644:
2370:.) Some facts of the projection matrix in this setting are summarized as follows:
4749:
4684:
4664:
4649:
4629:
4613:
4511:
4442:
4432:
4391:
4276:
4246:
3529:{\displaystyle \mathbf {P} =\mathbf {P} +\mathbf {P} {\big \mathbf {B} {\big ]},}
469:
5009:
4953:
4933:
4918:
4877:
4754:
4714:
4679:
4603:
4542:
4521:
4462:
4452:
4437:
4371:
4316:
4306:
4301:
4211:
3186:
Practical applications of the projection matrix in regression analysis include
2944:{\displaystyle \left(\mathbf {I} -\mathbf {P} \right)\mathbf {X} =\mathbf {0} }
2263:
827:
has its column space depicted as the green line. The projection of some vector
3198:, i.e. observations which have a large effect on the results of a regression.
3120:, for example, the hat matrix is in general neither symmetric nor idempotent.
5071:
5014:
4872:
4813:
4744:
4734:
4729:
4654:
4583:
4457:
4447:
4376:
4296:
4281:
4216:
4081:
Rao, C. Radhakrishna; Toutenburg, Helge; Shalabh; Heumann, Christian (2008).
3742:
3124:
1593:
1554:
4897:
4854:
4759:
4472:
4411:
4321:
4201:
4063:
3068:
2271:
101:
4739:
4709:
4477:
4311:
4181:
1042:{\displaystyle \mathbf {A} ^{\textsf {T}}(\mathbf {b} -\mathbf {Ax} )=0}
965:, and is one where we can draw a line orthogonal to the column space of
4790:
4251:
3935:
3926:
2741:
282:
27:
5024:
4598:
2418:{\displaystyle \mathbf {u} =(\mathbf {I} -\mathbf {P} )\mathbf {y} ,}
4064:"Proof that trace of 'hat' matrix in linear regression is rank of X"
3917:
896:
From the figure, it is clear that the closest point from the vector
4958:
4110:
1839:
Therefore, the projection matrix (and hat matrix) is given by
340:
can also be expressed compactly using the projection matrix:
216:{\displaystyle \mathbf {\hat {y}} =\mathbf {P} \mathbf {y} .}
1610:
When the weights for each observation are identical and the
4080:
3435:. Then the projection matrix can be decomposed as follows:
715:{\displaystyle \mathbf {\Sigma } =\sigma ^{2}\mathbf {I} }
1931:
3789:, which might be too large to fit into computer memory.
3428:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
2533:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
1563:
is a vector of unknown parameters to be estimated, and
503:{\displaystyle \mathbf {M} :=\mathbf {I} -\mathbf {P} }
3686:{\displaystyle \mathbf {M} =\mathbf {I} -\mathbf {P} }
3248:
3773:
3751:
3725:
3699:
3639:
3545:
3444:
3381:
3287:
3234:
3212:
3163:
3141:
3085:
3047:
2959:
2904:
2868:
2846:
2824:
2786:
2750:
2731:{\displaystyle \operatorname {rank} (\mathbf {P} )=r}
2704:
2691:{\displaystyle \operatorname {rank} (\mathbf {X} )=r}
2664:
2632:
2607:
2570:
2548:
2508:
2486:
2431:
2379:
2302:
2280:
2215:
2099:
1963:
1848:
1722:
1623:
1533:
1484:
1398:
1376:
1354:
1332:
1310:
1285:
1062:
996:
971:
946:
924:
902:
877:
855:
833:
811:
731:
688:
658:
560:
531:
478:
452:
349:
324:
291:
261:
255:
is usually pronounced "y-hat", the projection matrix
232:
184:
152:
130:
74:
39:
3895:Hoaglin, David C.; Welsch, Roy E. (February 1978).
4031:
3977:
3844:Applied Matrix Algebra in the Statistical Sciences
3781:
3759:
3733:
3707:
3685:
3625:
3528:
3427:
3367:
3273:
3220:
3179:, the projection matrix can be used to define the
3171:
3149:
3108:
3055:
3032:
2943:
2890:
2854:
2832:
2794:
2758:
2730:
2690:
2640:
2615:
2593:
2556:
2532:
2494:
2471:
2417:
2358:
2288:
2246:
2198:
2078:
1920:
1828:
1702:
1541:
1516:
1459:
1384:
1362:
1340:
1318:
1296:
1267:
1041:
979:
957:
932:
910:
885:
863:
841:
819:
784:
714:
666:
640:
539:
502:
460:
435:
332:
299:
269:
247:
215:
167:
138:
88:
53:
3953:
372:
239:
191:
159:
5069:
4038:. Cambridge: Harvard University Press. pp.
1614:are uncorrelated, the estimated parameters are
3118:locally weighted scatterplot smoothing (LOESS)
4126:
3894:
3890:
3888:
3518:
3490:
3375:. Similarly, define the residual operator as
3109:{\displaystyle \mathbf {P} ^{2}=\mathbf {P} }
2594:{\displaystyle \mathbf {P} ^{2}=\mathbf {P} }
100:(dependent variable values) to the vector of
3949:
3947:
3945:
310:
4004:
3281:. Define the hat or projection operator as
3116:. However, this is not always the case; in
2891:{\displaystyle \mathbf {PX} =\mathbf {X} ,}
4700:Fundamental (linear differential equation)
4133:
4119:
4089:(3rd ed.). Berlin: Springer. p.
3885:
3840:
3942:
3925:
3617:
3584:
3359:
3326:
3131:of the projection matrix is equal to the
3067:The projection matrix corresponding to a
2350:
2317:
2175:
2127:
2050:
2002:
1909:
1876:
1812:
1779:
1686:
1653:
1599:
1451:
1418:
1250:
1217:
1167:
1141:
1103:
1077:
1005:
604:
3897:"The Hat Matrix in Regression and ANOVA"
2254:, though now it is no longer symmetric.
800:
104:(or predicted values). It describes the
5005:Matrix representation of conic sections
4029:
3959:Statistical Models: Theory and Practice
3194:, which are concerned with identifying
1748:
1627:
1507:
1499:
680:independent and identically distributed
5070:
3767:without explicitly forming the matrix
1932:Weighted and generalized least squares
4114:
3980:Data Fitting in the Chemical Sciences
3975:
3201:
1326:, the projection matrix, which maps
248:{\displaystyle \mathbf {\hat {y}} }
168:{\displaystyle \mathbf {\hat {y}} }
146:and the vector of fitted values by
19:For the linear transformation, see
13:
4140:
667:{\displaystyle \mathbf {\Sigma } }
14:
5094:
4085:Linear Models and Generalizations
4005:Draper, N. R.; Smith, H. (1998).
5039:
3775:
3753:
3727:
3701:
3676:
3668:
3660:
3649:
3641:
3611:
3591:
3578:
3566:
3555:
3547:
3512:
3504:
3496:
3484:
3473:
3465:
3454:
3446:
3418:
3410:
3402:
3391:
3383:
3353:
3333:
3320:
3308:
3297:
3289:
3259:
3252:
3236:
3228:can be decomposed by columns as
3214:
3165:
3143:
3102:
3088:
3063:is unique for certain subspaces.
3049:
3023:
3010:
3002:
2992:
2984:
2974:
2966:
2937:
2929:
2919:
2911:
2881:
2873:
2870:
2848:
2826:
2788:
2780:zeros, while the eigenvalues of
2752:
2715:
2675:
2634:
2609:
2587:
2573:
2550:
2526:
2518:
2510:
2488:
2462:
2454:
2449:
2441:
2433:
2408:
2400:
2392:
2381:
2344:
2324:
2311:
2282:
2247:{\displaystyle H^{2}=H\cdot H=H}
2183:
2169:
2149:
2135:
2121:
2109:
2101:
2072:
2058:
2044:
2024:
2010:
1996:
1903:
1883:
1870:
1858:
1850:
1819:
1806:
1786:
1773:
1761:
1741:
1727:
1693:
1680:
1660:
1647:
1535:
1494:
1486:
1445:
1425:
1412:
1400:
1378:
1356:
1334:
1312:
1290:
1287:
1257:
1244:
1224:
1211:
1192:
1177:
1174:
1161:
1148:
1135:
1113:
1110:
1097:
1084:
1071:
1053:From there, one rearranges, so
1026:
1023:
1015:
999:
973:
951:
948:
926:
904:
879:
857:
835:
813:
763:
755:
740:
734:
708:
690:
660:
629:
621:
611:
593:
585:
569:
563:
533:
510:is sometimes referred to as the
496:
488:
480:
454:
426:
416:
408:
395:
390:
382:
369:
359:
351:
326:
293:
263:
236:
206:
201:
188:
156:
132:
79:
44:
4907:Used in science and engineering
2266:, the projection matrix is the
1470:
4150:Explicitly constrained entries
4074:
4056:
4023:
3998:
3969:
3861:
3841:Basilevsky, Alexander (2005).
3834:
3680:
3672:
3653:
3645:
3559:
3551:
3508:
3500:
3477:
3469:
3458:
3450:
3422:
3414:
3395:
3387:
3301:
3293:
2719:
2711:
2679:
2671:
2404:
2388:
2209:and again it may be seen that
1973:
1751:
1731:
1630:
1185:
1127:
1030:
1011:
315:The formula for the vector of
89:{\displaystyle (\mathbf {H} )}
83:
75:
54:{\displaystyle (\mathbf {P} )}
48:
40:
1:
4924:Fundamental (computer vision)
3827:
2257:
1297:{\displaystyle \mathbf {Ax} }
958:{\displaystyle \mathbf {Ax} }
115:
3817:Effective degrees of freedom
3782:{\displaystyle \mathbf {X} }
3760:{\displaystyle \mathbf {X} }
3734:{\displaystyle \mathbf {A} }
3708:{\displaystyle \mathbf {A} }
3221:{\displaystyle \mathbf {X} }
3181:effective degrees of freedom
3172:{\displaystyle \mathbf {y} }
3150:{\displaystyle \mathbf {X} }
3056:{\displaystyle \mathbf {P} }
2855:{\displaystyle \mathbf {P} }
2833:{\displaystyle \mathbf {X} }
2795:{\displaystyle \mathbf {M} }
2759:{\displaystyle \mathbf {P} }
2641:{\displaystyle \mathbf {X} }
2616:{\displaystyle \mathbf {M} }
2557:{\displaystyle \mathbf {P} }
2495:{\displaystyle \mathbf {P} }
2289:{\displaystyle \mathbf {X} }
1542:{\displaystyle \mathbf {X} }
1385:{\displaystyle \mathbf {A} }
1363:{\displaystyle \mathbf {x} }
1341:{\displaystyle \mathbf {b} }
1319:{\displaystyle \mathbf {A} }
980:{\displaystyle \mathbf {A} }
933:{\displaystyle \mathbf {A} }
911:{\displaystyle \mathbf {b} }
886:{\displaystyle \mathbf {x} }
864:{\displaystyle \mathbf {A} }
842:{\displaystyle \mathbf {b} }
820:{\displaystyle \mathbf {A} }
796:
540:{\displaystyle \mathbf {r} }
461:{\displaystyle \mathbf {I} }
333:{\displaystyle \mathbf {r} }
300:{\displaystyle \mathbf {y} }
270:{\displaystyle \mathbf {P} }
139:{\displaystyle \mathbf {y} }
61:, sometimes also called the
7:
4690:Duplication and elimination
4489:eigenvalues or eigenvectors
4007:Applied Regression Analysis
3847:. Dover. pp. 160–176.
3822:Mean and predicted response
3807:Projection (linear algebra)
3800:
21:Projection (linear algebra)
10:
5099:
4623:With specific applications
4252:Discrete Fourier Transform
3963:Cambridge University Press
3792:
3206:Suppose the design matrix
1935:
1603:
1304:is on the column space of
18:
5033:
4982:
4914:Cabibbo–Kobayashi–Maskawa
4906:
4852:
4788:
4622:
4541:Satisfying conditions on
4540:
4486:
4425:
4149:
4030:Amemiya, Takeshi (1985).
3905:The American Statistician
1942:Generalized least squares
1713:so the fitted values are
918:onto the column space of
849:onto the column space of
311:Application for residuals
3196:influential observations
2502:is symmetric, and so is
4272:Generalized permutation
2090:the hat matrix is thus
5046:Mathematics portal
3783:
3761:
3735:
3709:
3687:
3627:
3530:
3429:
3369:
3275:
3222:
3173:
3151:
3110:
3057:
3034:
2945:
2892:
2856:
2834:
2796:
2760:
2732:
2692:
2642:
2617:
2595:
2558:
2534:
2496:
2473:
2419:
2360:
2290:
2248:
2200:
2080:
1938:Weighted least squares
1922:
1830:
1704:
1606:Ordinary least squares
1600:Ordinary least squares
1543:
1518:
1461:
1386:
1364:
1342:
1320:
1298:
1269:
1043:
981:
959:
934:
912:
893:
887:
865:
843:
821:
786:
716:
668:
642:
541:
504:
462:
437:
334:
301:
271:
249:
217:
169:
140:
90:
55:
4034:Advanced Econometrics
3812:Studentized residuals
3784:
3762:
3736:
3710:
3688:
3628:
3531:
3430:
3370:
3276:
3223:
3174:
3152:
3111:
3058:
3035:
2946:
2893:
2857:
2835:
2797:
2761:
2733:
2693:
2643:
2618:
2596:
2559:
2535:
2497:
2474:
2420:
2361:
2291:
2274:of the design matrix
2268:orthogonal projection
2249:
2201:
2081:
1936:Further information:
1923:
1831:
1705:
1604:Further information:
1569:is the error vector.
1551:explanatory variables
1544:
1519:
1462:
1387:
1365:
1343:
1321:
1299:
1270:
1044:
982:
960:
935:
913:
888:
866:
844:
822:
804:
787:
717:
669:
643:
542:
512:residual maker matrix
505:
463:
438:
335:
302:
272:
250:
218:
170:
141:
96:, maps the vector of
91:
56:
16:Concept in statistics
3771:
3749:
3723:
3697:
3637:
3543:
3442:
3379:
3285:
3232:
3210:
3161:
3139:
3083:
3045:
2957:
2902:
2866:
2844:
2822:
2784:
2748:
2702:
2662:
2630:
2605:
2568:
2546:
2506:
2484:
2429:
2377:
2300:
2278:
2213:
2097:
1961:
1846:
1720:
1621:
1574:linear least squares
1531:
1482:
1396:
1374:
1352:
1330:
1308:
1283:
1060:
994:
969:
944:
922:
900:
875:
853:
831:
809:
729:
686:
656:
558:
529:
476:
450:
347:
322:
289:
259:
230:
182:
150:
128:
72:
37:
5078:Regression analysis
4995:Linear independence
4242:Diagonally dominant
3717:fixed effects model
2840:is invariant under
722:, this reduces to:
5000:Matrix exponential
4990:Jordan normal form
4824:Fisher information
4695:Euclidean distance
4609:Totally unimodular
3779:
3757:
3731:
3705:
3683:
3623:
3526:
3425:
3365:
3271:
3265:
3218:
3169:
3147:
3106:
3053:
3030:
2941:
2888:
2852:
2830:
2792:
2756:
2728:
2688:
2638:
2613:
2591:
2554:
2530:
2492:
2469:
2415:
2368:pseudoinverse of X
2356:
2286:
2244:
2196:
2076:
1918:
1826:
1700:
1582:regression splines
1539:
1514:
1457:
1382:
1360:
1338:
1316:
1294:
1265:
1263:
1039:
977:
955:
930:
908:
894:
883:
861:
839:
817:
782:
712:
664:
638:
537:
516:annihilator matrix
500:
458:
433:
330:
297:
267:
245:
213:
165:
136:
86:
51:
5065:
5064:
5057:Category:Matrices
4929:Fuzzy associative
4819:Doubly stochastic
4527:Positive-definite
4207:Block tridiagonal
4100:978-3-540-74226-5
4070:. April 13, 2017.
3976:Gans, P. (1992).
3955:David A. Freedman
3619:
3586:
3361:
3328:
3202:Blockwise formula
2352:
2319:
2177:
2129:
2052:
2004:
1982:
1976:
1950:of the errors is
1948:covariance matrix
1911:
1878:
1814:
1781:
1754:
1734:
1688:
1655:
1633:
1590:kernel regression
1578:smoothing splines
1453:
1420:
1279:Therefore, since
1252:
1219:
1169:
1143:
1105:
1079:
1007:
676:covariance matrix
606:
549:error propagation
525:of the residuals
523:covariance matrix
375:
242:
194:
162:
120:If the vector of
32:projection matrix
5090:
5052:List of matrices
5044:
5043:
5020:Row echelon form
4964:State transition
4893:Seidel adjacency
4775:Totally positive
4635:Alternating sign
4232:Complex Hadamard
4135:
4128:
4121:
4112:
4111:
4105:
4104:
4088:
4078:
4072:
4071:
4060:
4054:
4053:
4037:
4027:
4021:
4020:
4002:
3996:
3995:
3983:
3973:
3967:
3966:
3951:
3940:
3939:
3929:
3901:
3892:
3883:
3882:
3880:
3874:. Archived from
3873:
3865:
3859:
3858:
3838:
3788:
3786:
3785:
3780:
3778:
3766:
3764:
3763:
3758:
3756:
3740:
3738:
3737:
3732:
3730:
3714:
3712:
3711:
3706:
3704:
3692:
3690:
3689:
3684:
3679:
3671:
3663:
3652:
3644:
3632:
3630:
3629:
3624:
3622:
3621:
3620:
3614:
3608:
3607:
3599:
3595:
3594:
3589:
3588:
3587:
3581:
3569:
3558:
3550:
3535:
3533:
3532:
3527:
3522:
3521:
3515:
3507:
3499:
3494:
3493:
3487:
3476:
3468:
3457:
3449:
3434:
3432:
3431:
3426:
3421:
3413:
3405:
3394:
3386:
3374:
3372:
3371:
3366:
3364:
3363:
3362:
3356:
3350:
3349:
3341:
3337:
3336:
3331:
3330:
3329:
3323:
3311:
3300:
3292:
3280:
3278:
3277:
3272:
3270:
3269:
3262:
3255:
3239:
3227:
3225:
3224:
3219:
3217:
3178:
3176:
3175:
3170:
3168:
3156:
3154:
3153:
3148:
3146:
3115:
3113:
3112:
3107:
3105:
3097:
3096:
3091:
3062:
3060:
3059:
3054:
3052:
3039:
3037:
3036:
3031:
3026:
3018:
3014:
3013:
3005:
2995:
2987:
2982:
2978:
2977:
2969:
2950:
2948:
2947:
2942:
2940:
2932:
2927:
2923:
2922:
2914:
2897:
2895:
2894:
2889:
2884:
2876:
2861:
2859:
2858:
2853:
2851:
2839:
2837:
2836:
2831:
2829:
2811:
2801:
2799:
2798:
2793:
2791:
2779:
2765:
2763:
2762:
2757:
2755:
2737:
2735:
2734:
2729:
2718:
2697:
2695:
2694:
2689:
2678:
2657:
2647:
2645:
2644:
2639:
2637:
2622:
2620:
2619:
2614:
2612:
2600:
2598:
2597:
2592:
2590:
2582:
2581:
2576:
2563:
2561:
2560:
2555:
2553:
2539:
2537:
2536:
2531:
2529:
2521:
2513:
2501:
2499:
2498:
2493:
2491:
2478:
2476:
2475:
2470:
2465:
2457:
2452:
2444:
2436:
2424:
2422:
2421:
2416:
2411:
2403:
2395:
2384:
2365:
2363:
2362:
2357:
2355:
2354:
2353:
2347:
2341:
2340:
2332:
2328:
2327:
2322:
2321:
2320:
2314:
2295:
2293:
2292:
2287:
2285:
2253:
2251:
2250:
2245:
2225:
2224:
2205:
2203:
2202:
2197:
2195:
2194:
2186:
2180:
2179:
2178:
2172:
2166:
2165:
2157:
2153:
2152:
2147:
2146:
2138:
2132:
2131:
2130:
2124:
2112:
2104:
2085:
2083:
2082:
2077:
2075:
2070:
2069:
2061:
2055:
2054:
2053:
2047:
2041:
2040:
2032:
2028:
2027:
2022:
2021:
2013:
2007:
2006:
2005:
1999:
1984:
1983:
1980:
1978:
1977:
1972:
1967:
1927:
1925:
1924:
1919:
1914:
1913:
1912:
1906:
1900:
1899:
1891:
1887:
1886:
1881:
1880:
1879:
1873:
1861:
1853:
1835:
1833:
1832:
1827:
1822:
1817:
1816:
1815:
1809:
1803:
1802:
1794:
1790:
1789:
1784:
1783:
1782:
1776:
1764:
1756:
1755:
1747:
1744:
1736:
1735:
1730:
1725:
1709:
1707:
1706:
1701:
1696:
1691:
1690:
1689:
1683:
1677:
1676:
1668:
1664:
1663:
1658:
1657:
1656:
1650:
1635:
1634:
1626:
1594:linear filtering
1586:local regression
1548:
1546:
1545:
1540:
1538:
1523:
1521:
1520:
1515:
1510:
1502:
1497:
1489:
1466:
1464:
1463:
1458:
1456:
1455:
1454:
1448:
1442:
1441:
1433:
1429:
1428:
1423:
1422:
1421:
1415:
1403:
1391:
1389:
1388:
1383:
1381:
1369:
1367:
1366:
1361:
1359:
1347:
1345:
1344:
1339:
1337:
1325:
1323:
1322:
1317:
1315:
1303:
1301:
1300:
1295:
1293:
1274:
1272:
1271:
1266:
1264:
1260:
1255:
1254:
1253:
1247:
1241:
1240:
1232:
1228:
1227:
1222:
1221:
1220:
1214:
1195:
1189:
1180:
1172:
1171:
1170:
1164:
1151:
1146:
1145:
1144:
1138:
1131:
1116:
1108:
1107:
1106:
1100:
1087:
1082:
1081:
1080:
1074:
1067:
1066:
1048:
1046:
1045:
1040:
1029:
1018:
1010:
1009:
1008:
1002:
986:
984:
983:
978:
976:
964:
962:
961:
956:
954:
939:
937:
936:
931:
929:
917:
915:
914:
909:
907:
892:
890:
889:
884:
882:
870:
868:
867:
862:
860:
848:
846:
845:
840:
838:
826:
824:
823:
818:
816:
791:
789:
788:
783:
781:
780:
771:
767:
766:
758:
745:
744:
743:
737:
721:
719:
718:
713:
711:
706:
705:
693:
682:errors in which
673:
671:
670:
665:
663:
647:
645:
644:
639:
637:
633:
632:
624:
614:
609:
608:
607:
601:
597:
596:
588:
574:
573:
572:
566:
546:
544:
543:
538:
536:
509:
507:
506:
501:
499:
491:
483:
467:
465:
464:
459:
457:
442:
440:
439:
434:
429:
424:
420:
419:
411:
398:
393:
385:
377:
376:
368:
362:
354:
339:
337:
336:
331:
329:
306:
304:
303:
298:
296:
276:
274:
273:
268:
266:
254:
252:
251:
246:
244:
243:
235:
222:
220:
219:
214:
209:
204:
196:
195:
187:
174:
172:
171:
166:
164:
163:
155:
145:
143:
142:
137:
135:
95:
93:
92:
87:
82:
63:influence matrix
60:
58:
57:
52:
47:
5098:
5097:
5093:
5092:
5091:
5089:
5088:
5087:
5068:
5067:
5066:
5061:
5038:
5029:
4978:
4902:
4848:
4784:
4618:
4536:
4482:
4421:
4222:Centrosymmetric
4145:
4139:
4109:
4108:
4101:
4079:
4075:
4062:
4061:
4057:
4050:
4028:
4024:
4017:
4003:
3999:
3992:
3974:
3970:
3952:
3943:
3918:10.2307/2683469
3899:
3893:
3886:
3878:
3871:
3867:
3866:
3862:
3855:
3839:
3835:
3830:
3803:
3795:
3774:
3772:
3769:
3768:
3752:
3750:
3747:
3746:
3726:
3724:
3721:
3720:
3700:
3698:
3695:
3694:
3675:
3667:
3659:
3648:
3640:
3638:
3635:
3634:
3616:
3615:
3610:
3609:
3600:
3590:
3583:
3582:
3577:
3576:
3575:
3571:
3570:
3565:
3554:
3546:
3544:
3541:
3540:
3517:
3516:
3511:
3503:
3495:
3489:
3488:
3483:
3472:
3464:
3453:
3445:
3443:
3440:
3439:
3417:
3409:
3401:
3390:
3382:
3380:
3377:
3376:
3358:
3357:
3352:
3351:
3342:
3332:
3325:
3324:
3319:
3318:
3317:
3313:
3312:
3307:
3296:
3288:
3286:
3283:
3282:
3264:
3263:
3258:
3256:
3251:
3244:
3243:
3235:
3233:
3230:
3229:
3213:
3211:
3208:
3207:
3204:
3192:Cook's distance
3164:
3162:
3159:
3158:
3142:
3140:
3137:
3136:
3101:
3092:
3087:
3086:
3084:
3081:
3080:
3048:
3046:
3043:
3042:
3022:
3009:
3001:
3000:
2996:
2991:
2983:
2973:
2965:
2964:
2960:
2958:
2955:
2954:
2936:
2928:
2918:
2910:
2909:
2905:
2903:
2900:
2899:
2880:
2869:
2867:
2864:
2863:
2847:
2845:
2842:
2841:
2825:
2823:
2820:
2819:
2803:
2787:
2785:
2782:
2781:
2771:
2751:
2749:
2746:
2745:
2714:
2703:
2700:
2699:
2674:
2663:
2660:
2659:
2649:
2633:
2631:
2628:
2627:
2608:
2606:
2603:
2602:
2586:
2577:
2572:
2571:
2569:
2566:
2565:
2564:is idempotent:
2549:
2547:
2544:
2543:
2525:
2517:
2509:
2507:
2504:
2503:
2487:
2485:
2482:
2481:
2461:
2453:
2448:
2440:
2432:
2430:
2427:
2426:
2407:
2399:
2391:
2380:
2378:
2375:
2374:
2349:
2348:
2343:
2342:
2333:
2323:
2316:
2315:
2310:
2309:
2308:
2304:
2303:
2301:
2298:
2297:
2281:
2279:
2276:
2275:
2260:
2220:
2216:
2214:
2211:
2210:
2187:
2182:
2181:
2174:
2173:
2168:
2167:
2158:
2148:
2139:
2134:
2133:
2126:
2125:
2120:
2119:
2118:
2114:
2113:
2108:
2100:
2098:
2095:
2094:
2071:
2062:
2057:
2056:
2049:
2048:
2043:
2042:
2033:
2023:
2014:
2009:
2008:
2001:
2000:
1995:
1994:
1993:
1989:
1988:
1979:
1968:
1966:
1965:
1964:
1962:
1959:
1958:
1944:
1934:
1908:
1907:
1902:
1901:
1892:
1882:
1875:
1874:
1869:
1868:
1867:
1863:
1862:
1857:
1849:
1847:
1844:
1843:
1818:
1811:
1810:
1805:
1804:
1795:
1785:
1778:
1777:
1772:
1771:
1770:
1766:
1765:
1760:
1746:
1745:
1740:
1726:
1724:
1723:
1721:
1718:
1717:
1692:
1685:
1684:
1679:
1678:
1669:
1659:
1652:
1651:
1646:
1645:
1644:
1640:
1639:
1625:
1624:
1622:
1619:
1618:
1608:
1602:
1549:is a matrix of
1534:
1532:
1529:
1528:
1506:
1498:
1493:
1485:
1483:
1480:
1479:
1473:
1450:
1449:
1444:
1443:
1434:
1424:
1417:
1416:
1411:
1410:
1409:
1405:
1404:
1399:
1397:
1394:
1393:
1377:
1375:
1372:
1371:
1355:
1353:
1350:
1349:
1333:
1331:
1328:
1327:
1311:
1309:
1306:
1305:
1286:
1284:
1281:
1280:
1262:
1261:
1256:
1249:
1248:
1243:
1242:
1233:
1223:
1216:
1215:
1210:
1209:
1208:
1204:
1203:
1196:
1191:
1188:
1182:
1181:
1173:
1166:
1165:
1160:
1159:
1152:
1147:
1140:
1139:
1134:
1133:
1130:
1124:
1123:
1109:
1102:
1101:
1096:
1095:
1088:
1083:
1076:
1075:
1070:
1069:
1063:
1061:
1058:
1057:
1022:
1014:
1004:
1003:
998:
997:
995:
992:
991:
972:
970:
967:
966:
947:
945:
942:
941:
925:
923:
920:
919:
903:
901:
898:
897:
878:
876:
873:
872:
856:
854:
851:
850:
834:
832:
829:
828:
812:
810:
807:
806:
799:
776:
772:
762:
754:
753:
749:
739:
738:
733:
732:
730:
727:
726:
707:
701:
697:
689:
687:
684:
683:
659:
657:
654:
653:
628:
620:
619:
615:
610:
603:
602:
592:
584:
583:
579:
578:
568:
567:
562:
561:
559:
556:
555:
532:
530:
527:
526:
495:
487:
479:
477:
474:
473:
470:identity matrix
453:
451:
448:
447:
425:
415:
407:
406:
402:
394:
389:
381:
367:
366:
358:
350:
348:
345:
344:
325:
323:
320:
319:
313:
292:
290:
287:
286:
262:
260:
257:
256:
234:
233:
231:
228:
227:
205:
200:
186:
185:
183:
180:
179:
154:
153:
151:
148:
147:
131:
129:
126:
125:
122:response values
118:
98:response values
78:
73:
70:
69:
43:
38:
35:
34:
24:
17:
12:
11:
5:
5096:
5086:
5085:
5080:
5063:
5062:
5060:
5059:
5054:
5049:
5034:
5031:
5030:
5028:
5027:
5022:
5017:
5012:
5010:Perfect matrix
5007:
5002:
4997:
4992:
4986:
4984:
4980:
4979:
4977:
4976:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4910:
4908:
4904:
4903:
4901:
4900:
4895:
4890:
4885:
4880:
4875:
4870:
4865:
4859:
4857:
4850:
4849:
4847:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4811:
4806:
4801:
4795:
4793:
4786:
4785:
4783:
4782:
4780:Transformation
4777:
4772:
4767:
4762:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4722:
4717:
4712:
4707:
4702:
4697:
4692:
4687:
4682:
4677:
4672:
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4626:
4624:
4620:
4619:
4617:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4576:
4571:
4566:
4557:
4551:
4549:
4538:
4537:
4535:
4534:
4529:
4524:
4519:
4517:Diagonalizable
4514:
4509:
4504:
4499:
4493:
4491:
4487:Conditions on
4484:
4483:
4481:
4480:
4475:
4470:
4465:
4460:
4455:
4450:
4445:
4440:
4435:
4429:
4427:
4423:
4422:
4420:
4419:
4414:
4409:
4404:
4399:
4394:
4389:
4384:
4379:
4374:
4369:
4367:Skew-symmetric
4364:
4362:Skew-Hermitian
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4197:Block-diagonal
4194:
4189:
4184:
4179:
4174:
4172:Anti-symmetric
4169:
4167:Anti-Hermitian
4164:
4159:
4153:
4151:
4147:
4146:
4138:
4137:
4130:
4123:
4115:
4107:
4106:
4099:
4073:
4068:Stack Exchange
4055:
4048:
4022:
4015:
3997:
3990:
3968:
3941:
3884:
3881:on 2014-09-03.
3860:
3853:
3832:
3831:
3829:
3826:
3825:
3824:
3819:
3814:
3809:
3802:
3799:
3794:
3791:
3777:
3755:
3729:
3703:
3682:
3678:
3674:
3670:
3666:
3662:
3658:
3655:
3651:
3647:
3643:
3613:
3606:
3603:
3598:
3593:
3580:
3574:
3568:
3564:
3561:
3557:
3553:
3549:
3537:
3536:
3525:
3520:
3514:
3510:
3506:
3502:
3498:
3492:
3486:
3482:
3479:
3475:
3471:
3467:
3463:
3460:
3456:
3452:
3448:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3397:
3393:
3389:
3385:
3355:
3348:
3345:
3340:
3335:
3322:
3316:
3310:
3306:
3303:
3299:
3295:
3291:
3268:
3261:
3257:
3254:
3250:
3249:
3247:
3242:
3238:
3216:
3203:
3200:
3183:of the model.
3167:
3145:
3104:
3100:
3095:
3090:
3065:
3064:
3051:
3040:
3029:
3025:
3021:
3017:
3012:
3008:
3004:
2999:
2994:
2990:
2986:
2981:
2976:
2972:
2968:
2963:
2952:
2939:
2935:
2931:
2926:
2921:
2917:
2913:
2908:
2887:
2883:
2879:
2875:
2872:
2850:
2828:
2817:
2790:
2754:
2738:
2727:
2724:
2721:
2717:
2713:
2710:
2707:
2687:
2684:
2681:
2677:
2673:
2670:
2667:
2636:
2624:
2611:
2589:
2585:
2580:
2575:
2552:
2541:
2528:
2524:
2520:
2516:
2512:
2490:
2479:
2468:
2464:
2460:
2456:
2451:
2447:
2443:
2439:
2435:
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2387:
2383:
2346:
2339:
2336:
2331:
2326:
2313:
2307:
2284:
2264:linear algebra
2259:
2256:
2243:
2240:
2237:
2234:
2231:
2228:
2223:
2219:
2207:
2206:
2193:
2190:
2185:
2171:
2164:
2161:
2156:
2151:
2145:
2142:
2137:
2123:
2117:
2111:
2107:
2103:
2088:
2087:
2074:
2068:
2065:
2060:
2046:
2039:
2036:
2031:
2026:
2020:
2017:
2012:
1998:
1992:
1987:
1975:
1971:
1954:. Then since
1933:
1930:
1929:
1928:
1917:
1905:
1898:
1895:
1890:
1885:
1872:
1866:
1860:
1856:
1852:
1837:
1836:
1825:
1821:
1808:
1801:
1798:
1793:
1788:
1775:
1769:
1763:
1759:
1753:
1750:
1743:
1739:
1733:
1729:
1711:
1710:
1699:
1695:
1682:
1675:
1672:
1667:
1662:
1649:
1643:
1638:
1632:
1629:
1601:
1598:
1537:
1525:
1524:
1513:
1509:
1505:
1501:
1496:
1492:
1488:
1472:
1469:
1447:
1440:
1437:
1432:
1427:
1414:
1408:
1402:
1380:
1358:
1336:
1314:
1292:
1289:
1277:
1276:
1259:
1246:
1239:
1236:
1231:
1226:
1213:
1207:
1202:
1199:
1197:
1194:
1190:
1187:
1184:
1183:
1179:
1176:
1163:
1158:
1155:
1153:
1150:
1137:
1132:
1129:
1126:
1125:
1122:
1119:
1115:
1112:
1099:
1094:
1091:
1089:
1086:
1073:
1068:
1065:
1051:
1050:
1038:
1035:
1032:
1028:
1025:
1021:
1017:
1013:
1001:
975:
953:
950:
928:
906:
881:
871:is the vector
859:
837:
815:
798:
795:
794:
793:
779:
775:
770:
765:
761:
757:
752:
748:
742:
736:
710:
704:
700:
696:
692:
662:
650:
649:
636:
631:
627:
623:
618:
613:
600:
595:
591:
587:
582:
577:
571:
565:
535:
498:
494:
490:
486:
482:
456:
444:
443:
432:
428:
423:
418:
414:
410:
405:
401:
397:
392:
388:
384:
380:
374:
371:
365:
361:
357:
353:
328:
312:
309:
295:
281:as it "puts a
277:is also named
265:
241:
238:
224:
223:
212:
208:
203:
199:
193:
190:
161:
158:
134:
124:is denoted by
117:
114:
85:
81:
77:
50:
46:
42:
15:
9:
6:
4:
3:
2:
5095:
5084:
5081:
5079:
5076:
5075:
5073:
5058:
5055:
5053:
5050:
5048:
5047:
5042:
5036:
5035:
5032:
5026:
5023:
5021:
5018:
5016:
5015:Pseudoinverse
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4987:
4985:
4983:Related terms
4981:
4975:
4974:Z (chemistry)
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4911:
4909:
4905:
4899:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4860:
4858:
4856:
4851:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4796:
4794:
4792:
4787:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4621:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4561:
4558:
4556:
4553:
4552:
4550:
4548:
4544:
4539:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4494:
4492:
4490:
4485:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4430:
4428:
4424:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4332:Pentadiagonal
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4162:Anti-diagonal
4160:
4158:
4155:
4154:
4152:
4148:
4143:
4136:
4131:
4129:
4124:
4122:
4117:
4116:
4113:
4102:
4096:
4092:
4087:
4086:
4077:
4069:
4065:
4059:
4051:
4049:0-674-00560-0
4045:
4041:
4036:
4035:
4026:
4018:
4016:0-471-17082-8
4012:
4008:
4001:
3993:
3991:0-471-93412-7
3987:
3982:
3981:
3972:
3964:
3960:
3956:
3950:
3948:
3946:
3937:
3933:
3928:
3923:
3919:
3915:
3911:
3907:
3906:
3898:
3891:
3889:
3877:
3870:
3864:
3856:
3854:0-486-44538-0
3850:
3846:
3845:
3837:
3833:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3804:
3798:
3790:
3744:
3743:sparse matrix
3718:
3664:
3656:
3604:
3601:
3596:
3572:
3562:
3539:where, e.g.,
3523:
3480:
3461:
3438:
3437:
3436:
3406:
3398:
3346:
3343:
3338:
3314:
3304:
3266:
3245:
3240:
3199:
3197:
3193:
3189:
3184:
3182:
3134:
3130:
3126:
3125:linear models
3121:
3119:
3098:
3093:
3078:
3074:
3070:
3041:
3027:
3019:
3015:
3006:
2997:
2988:
2979:
2970:
2961:
2953:
2933:
2924:
2915:
2906:
2885:
2877:
2818:
2815:
2810:
2806:
2778:
2774:
2769:
2743:
2739:
2725:
2722:
2708:
2705:
2685:
2682:
2668:
2665:
2656:
2652:
2625:
2583:
2578:
2542:
2522:
2514:
2480:
2466:
2458:
2445:
2437:
2412:
2396:
2385:
2373:
2372:
2371:
2369:
2337:
2334:
2329:
2305:
2296:. (Note that
2273:
2269:
2265:
2255:
2241:
2238:
2235:
2232:
2229:
2226:
2221:
2217:
2191:
2188:
2162:
2159:
2154:
2143:
2140:
2115:
2105:
2093:
2092:
2091:
2066:
2063:
2037:
2034:
2029:
2018:
2015:
1990:
1985:
1969:
1957:
1956:
1955:
1953:
1949:
1943:
1939:
1915:
1896:
1893:
1888:
1864:
1854:
1842:
1841:
1840:
1823:
1799:
1796:
1791:
1767:
1757:
1737:
1716:
1715:
1714:
1697:
1673:
1670:
1665:
1641:
1636:
1617:
1616:
1615:
1613:
1607:
1597:
1595:
1591:
1587:
1583:
1579:
1575:
1570:
1568:
1567:
1562:
1561:
1556:
1555:design matrix
1552:
1511:
1503:
1490:
1478:
1477:
1476:
1468:
1438:
1435:
1430:
1406:
1237:
1234:
1229:
1205:
1200:
1198:
1156:
1154:
1120:
1117:
1092:
1090:
1056:
1055:
1054:
1036:
1033:
1019:
990:
989:
988:
803:
777:
773:
768:
759:
750:
746:
725:
724:
723:
702:
698:
694:
681:
677:
634:
625:
616:
598:
589:
580:
575:
554:
553:
552:
550:
524:
519:
517:
513:
492:
484:
472:. The matrix
471:
430:
421:
412:
403:
399:
386:
378:
363:
355:
343:
342:
341:
318:
308:
284:
280:
210:
197:
178:
177:
176:
123:
113:
111:
107:
103:
102:fitted values
99:
68:
64:
33:
29:
22:
5037:
4969:Substitution
4855:graph theory
4828:
4352:Quaternionic
4342:Persymmetric
4084:
4076:
4067:
4058:
4033:
4025:
4006:
4000:
3979:
3971:
3958:
3912:(1): 17–22.
3909:
3903:
3876:the original
3863:
3843:
3836:
3796:
3538:
3205:
3185:
3122:
3069:linear model
3066:
2813:
2808:
2804:
2776:
2772:
2767:
2658:matrix with
2654:
2650:
2601:, and so is
2272:column space
2261:
2208:
2089:
1951:
1945:
1838:
1712:
1609:
1571:
1565:
1564:
1559:
1558:
1526:
1474:
1471:Linear model
1278:
1052:
895:
651:
520:
515:
511:
445:
314:
278:
225:
119:
66:
62:
31:
25:
4944:Hamiltonian
4868:Biadjacency
4804:Correlation
4720:Householder
4670:Commutation
4407:Vandermonde
4402:Tridiagonal
4337:Permutation
4327:Nonnegative
4312:Matrix unit
4192:Bisymmetric
3927:1721.1/1920
3741:is a large
3079:, that is,
2802:consist of
2766:consist of
2742:eigenvalues
5072:Categories
4844:Transition
4839:Stochastic
4809:Covariance
4791:statistics
4770:Symplectic
4765:Similarity
4594:Unimodular
4589:Orthogonal
4574:Involutory
4569:Invertible
4564:Projection
4560:Idempotent
4502:Convergent
4397:Triangular
4347:Polynomial
4292:Hessenberg
4262:Equivalent
4257:Elementary
4237:Copositive
4227:Conference
4187:Bidiagonal
3828:References
3077:idempotent
2258:Properties
805:A matrix,
279:hat matrix
116:Definition
67:hat matrix
28:statistics
5025:Wronskian
4949:Irregular
4939:Gell-Mann
4888:Laplacian
4883:Incidence
4863:Adjacency
4834:Precision
4799:Centering
4705:Generator
4675:Confusion
4660:Circulant
4640:Augmented
4599:Unipotent
4579:Nilpotent
4555:Congruent
4532:Stieltjes
4507:Defective
4497:Companion
4468:Redheffer
4387:Symmetric
4382:Sylvester
4357:Signature
4287:Hermitian
4267:Frobenius
4177:Arrowhead
4157:Alternant
4009:. Wiley.
3984:. Wiley.
3665:−
3602:−
3407:−
3344:−
3073:symmetric
3007:−
2971:−
2916:−
2812:ones and
2770:ones and
2709:
2669:
2523:−
2459:⊥
2446:−
2397:−
2335:−
2270:onto the
2233:⋅
2189:−
2184:Σ
2160:−
2141:−
2136:Σ
2064:−
2059:Σ
2035:−
2016:−
2011:Σ
1974:^
1970:β
1894:−
1797:−
1752:^
1749:β
1732:^
1671:−
1631:^
1628:β
1508:ε
1500:β
1436:−
1235:−
1186:⇒
1128:⇒
1093:−
1020:−
797:Intuition
774:σ
760:−
735:Σ
699:σ
691:Σ
661:Σ
626:−
612:Σ
590:−
564:Σ
551:, equals
493:−
413:−
387:−
373:^
364:−
317:residuals
240:^
192:^
160:^
110:leverages
106:influence
5083:Matrices
4853:Used in
4789:Used in
4750:Rotation
4725:Jacobian
4685:Distance
4665:Cofactor
4650:Carleman
4630:Adjugate
4614:Weighing
4547:inverses
4543:products
4512:Definite
4443:Identity
4433:Exchange
4426:Constant
4392:Toeplitz
4277:Hadamard
4247:Diagonal
3957:(2009).
3801:See also
3719:, where
3188:leverage
2862: :
1370:is just
4954:Overlap
4919:Density
4878:Edmonds
4755:Seifert
4715:Hessian
4680:Coxeter
4604:Unitary
4522:Hurwitz
4453:Of ones
4438:Hilbert
4372:Skyline
4317:Metzler
4307:Logical
4302:Integer
4212:Boolean
4144:classes
3936:2683469
3793:History
2698:, then
2366:is the
674:is the
514:or the
468:is the
4873:Degree
4814:Design
4745:Random
4735:Payoff
4730:Moment
4655:Cartan
4645:Bézout
4584:Normal
4458:Pascal
4448:Lehmer
4377:Sparse
4297:Hollow
4282:Hankel
4217:Cauchy
4142:Matrix
4097:
4046:
4042:–461.
4013:
3988:
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3851:
3127:, the
2898:hence
2816:zeros.
2648:is an
1612:errors
1592:, and
1527:where
652:where
446:where
30:, the
4934:Gamma
4898:Tutte
4760:Shear
4473:Shift
4463:Pauli
4412:Walsh
4322:Moore
4202:Block
3932:JSTOR
3900:(PDF)
3879:(PDF)
3872:(PDF)
3129:trace
1553:(the
1392:, or
1348:onto
940:, is
547:, by
4740:Pick
4710:Gram
4478:Zero
4182:Band
4095:ISBN
4044:ISBN
4011:ISBN
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3849:ISBN
3633:and
3190:and
3133:rank
3123:For
3075:and
2740:The
2706:rank
2666:rank
2425:and
1940:and
521:The
4829:Hat
4562:or
4545:or
4091:323
4040:460
3922:hdl
3914:doi
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3071:is
2744:of
2626:If
1981:GLS
1557:),
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307:".
285:on
283:hat
226:As
65:or
26:In
5074::
4093:.
4066:.
3961:.
3944:^
3930:.
3920:.
3910:32
3908:.
3902:.
3887:^
3399::=
3305::=
2807:−
2775:−
2653:×
2515::=
1855::=
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1584:,
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1467:.
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4959:S
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3677:A
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3650:A
3646:[
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3556:A
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3509:]
3505:A
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1997:X
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1952:Σ
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