3567:
1788:
1262:
1367:
1783:{\displaystyle {\begin{pmatrix}3&2&2\\1&1&0\\1&0&1\end{pmatrix}}{\begin{pmatrix}a&0&0\\0&b&0\\0&0&c\end{pmatrix}}{\begin{pmatrix}3&2&2\\1&1&0\\1&0&1\end{pmatrix}}^{-1}={\begin{pmatrix}-3a+2b+2c&6a-2b-4c&6a-4b-2c\\b-a&a+(a-b)&2(a-b)\\c-a&2(a-c)&a+(a-c)\end{pmatrix}}}
951:
3165:
2089:. This is however just a happy coincidence; if working through the steps of the proof one finds that it in each eigenvector is the first element that is the largest (every eigenspace is closer to the first axis than to any other axis), so the theorem only promises that the disc for row 1 (whose radius can be twice the
3409:
The proof given above is arguably (in)correct...... There are two types of continuity concerning eigenvalues: (1) each individual eigenvalue is a usual continuous function (such a representation does exist on a real interval but may not exist on a complex domain), (2) eigenvalues are continuous as a
1321:, the eigenvalues of the matrix cannot be "far from" the diagonal entries of the matrix. Therefore, by reducing the norms of off-diagonal entries one can attempt to approximate the eigenvalues of the matrix. Of course, diagonal entries may change in the process of minimizing off-diagonal entries.
3169:
The union of the first 3 disks does not intersect the last 2, but the matrix has only 2 eigenvectors, e1,e4, and therefore only 2 eigenvalues, demonstrating that theorem is false in its formulation. The demonstration of the shows only that eigenvalues are distinct, however any affirmation about
1257:{\displaystyle \left|\lambda -a_{ii}\right|=\left|\sum _{j\neq i}{\frac {a_{ij}x_{j}}{x_{i}}}\right|\leq \sum _{j\neq i}\left|{\frac {a_{ij}x_{j}}{x_{i}}}\right|=\sum _{j\neq i}\left|a_{ij}\right|{\frac {\left|x_{j}\right|}{\left|x_{i}\right|}}\leq \sum _{j\neq i}\left|a_{ij}\right|=R_{i}.}
3711:
4081:. This means that most of the matrix is in the diagonal, which explains why the eigenvalues are so close to the centers of the circles, and the estimates are very good. For a random matrix, we would expect the eigenvalues to be substantially further from the centers of the circles.
3414:^n under permutation equivalence with induced metric). Whichever continuity is used in a proof of the Gerschgorin disk theorem, it should be justified that the sum of algebraic multiplicities of eigenvalues remains unchanged on each connected region. A proof using the
3006:
2227:
2160:
3570:
This diagram shows the discs in yellow derived for the eigenvalues. The first two disks overlap and their union contains two eigenvalues. The third and fourth disks are disjoint from the others and contain one eigenvalue
2007:
1955:
1903:
3578:
939:
3914:
3371:
could merge with other eigenvalue(s) or appeared from a splitting of previous eigenvalue. This may confuse people and questioning the concept of continuous. However, when viewing from the space of eigenvalue set
869:
4178:
294:
357:
736:
4079:
3795:
2101:
If one of the discs is disjoint from the others then it contains exactly one eigenvalue. If however it meets another disc it is possible that it contains no eigenvalue (for example,
3160:{\displaystyle {\begin{bmatrix}5&1&0&0&0\\0&5&1&0&0\\0&0&5&0&0\\0&0&0&1&1\\0&0&0&0&1\end{bmatrix}}}
168:
2925:
3478:
is accurate to three decimal places. For very high condition numbers, even very small errors due to rounding can be magnified to such an extent that the result is meaningless.
3399:
3234:
1359:
504:
2961:
2877:
2343:
3275:
660:
3369:
3308:
3201:
2990:
2799:
2667:
3990:
2770:
589:
94:
3952:
530:
2838:
2566:
2424:
769:
391:
124:
2536:
628:
418:
195:
3337:
2729:
2696:
2634:
2595:
2494:
2453:
2386:
2366:
2087:
2067:
2047:
2027:
1851:
1831:
1811:
550:
449:
219:
65:
2165:
2104:
39:
in 1931. Gershgorin's name has been transliterated in several different ways, including GerĹĄgorin, Gerschgorin, Gershgorin, Hershhorn, and
Hirschhorn.
3281:. This proves the roots as a whole is a continuous function of its coefficients. Since composition of continuous functions is again continuous, the
3706:{\displaystyle A={\begin{bmatrix}10&1&0&1\\0.2&8&0.2&0.2\\1&1&2&1\\-1&-1&-1&-11\\\end{bmatrix}}.}
1960:
1908:
1856:
3410:
whole in the topological sense (a mapping from the matrix space with metric induced by a norm to unordered tuples, i.e., the quotient space of
881:
3806:
1309:, the Gershgorin discs coincide with the spectrum. Conversely, if the Gershgorin discs coincide with the spectrum, the matrix is diagonal.
3919:
Note that we can improve the accuracy of the last two discs by applying the formula to the corresponding columns of the matrix, obtaining
777:
3000:
Remarks: It is necessary to count the eigenvalues with respect to their algebraic multiplicities. Here is a counter-example :
4162:
227:
1361:, and each expresses a bound on precisely those eigenvalues whose eigenspaces are closest to one particular axis. In the matrix
1317:
One way to interpret this theorem is that if the off-diagonal entries of a square matrix over the complex numbers have small
2368:, and show that if any eigenvalue moves from one of the unions to the other, then it must be outside all the discs for some
302:
672:
3236:) is continuous function of its coefficients. Note that the inverse map that maps roots to coefficients is described by
3524:
would be nice but finding the inverse of a matrix is something we want to avoid because of the computational expense.
4266:
4223:
4158:
4002:
3731:
28:
4319:
4091:
3204:
4304:
4190:
129:
36:
2882:
4151:
3996:
3375:
3210:
1335:
1299:
while recognizing that the eigenvalues of the transpose are the same as those of the original matrix.
454:
4122:
3241:
2930:
2843:
2286:
4117:
4097:
3247:
3725:
as the center for the disc. We then take the remaining elements in the row and apply the formula
1328:
claim that there is one disc for each eigenvalue; if anything, the discs rather correspond to the
633:
4324:
3345:
3284:
3177:
2966:
2775:
2643:
3957:
2740:
4329:
4127:
555:
73:
4163:
https://www.cambridge.org/ca/academic/subjects/mathematics/algebra/matrix-analysis-2nd-edition
3922:
515:
3422:
requires no eigenvalue continuity of any kind. For a brief discussion and clarification, see.
4297:
2808:
2545:
2394:
744:
366:
99:
2503:
606:
396:
173:
32:
3313:
2705:
2672:
2610:
2571:
2470:
2429:
8:
4254:
3237:
875:
3401:, the trajectory is still a continuous curve although not necessarily smooth everywhere.
3551:
lies within a known area and so we can form a rough estimate of how good our choice of
3459:
3415:
2371:
2351:
2222:{\displaystyle A=\left({\begin{smallmatrix}1&-2\\1&-1\end{smallmatrix}}\right)}
2072:
2052:
2032:
2012:
1836:
1816:
1796:
1318:
535:
434:
360:
204:
50:
4230:
4262:
4219:
4154:
2155:{\displaystyle A=\left({\begin{smallmatrix}0&1\\4&0\end{smallmatrix}}\right)}
4237:
4211:
4107:
3452:
3419:
2459:, thus the centers of the Gershgorin circles are the same, however their radii are
4283:
3566:
3486:
1306:
3995:
The eigenvalues are -10.870, 1.906, 10.046, 7.918. Note that this is a (column)
3547:
should all be close to 1. By the
Gershgorin circle theorem, every eigenvalue of
3431:
The
Gershgorin circle theorem is useful in solving matrix equations of the form
2181:
2120:
1970:
1918:
1866:
4112:
4102:
3517:
600:
198:
68:
4194:
3458:
In this kind of problem, the error in the final result is usually of the same
4313:
4250:
3170:
number of them is something that does not fit, and this is a counterexample.
2002:{\displaystyle \left({\begin{smallmatrix}2\\0\\1\end{smallmatrix}}\right)}
1950:{\displaystyle \left({\begin{smallmatrix}2\\1\\0\end{smallmatrix}}\right)}
1898:{\displaystyle \left({\begin{smallmatrix}3\\1\\1\end{smallmatrix}}\right)}
20:
4288:
3540:
3462:
as the error in the initial data multiplied by the condition number of
428:
2265:
when the eigenvalues are counted with their algebraic multiplicities.
934:{\displaystyle {\frac {\left|x_{j}\right|}{\left|x_{i}\right|}}\leq 1}
3909:{\displaystyle D(10,2),\;D(8,0.6),\;D(2,3),\;{\text{and}}\;D(-11,3).}
2276:
be the diagonal matrix with entries equal to the diagonal entries of
3278:
2229:). In the general case the theorem can be strengthened as follows:
3563:
Use the
Gershgorin circle theorem to estimate the eigenvalues of:
2963:
lies outside the
Gershgorin discs, which is impossible. Therefore
864:{\displaystyle \sum _{j\neq i}a_{ij}x_{j}=(\lambda -a_{ii})x_{i}.}
3207:. It is sufficient to show that the roots (as a point in space
2538:. The discs are closed, so the distance of the two unions for
3716:
Starting with row one, we take the element on the diagonal,
2348:
We will use the fact that the eigenvalues are continuous in
3470:
is known to six decimal places and the condition number of
2463:
times that of A. Therefore, the union of the corresponding
289:{\displaystyle R_{i}=\sum _{j\neq {i}}\left|a_{ij}\right|.}
4261:, Baltimore: Johns Hopkins University Press, p. 320,
4005:
3593:
3015:
2093:
of the other two radii) covers all three eigenvalues.
1578:
1503:
1439:
1376:
3960:
3925:
3809:
3734:
3581:
3378:
3348:
3316:
3287:
3250:
3213:
3180:
3009:
2969:
2933:
2885:
2846:
2811:
2778:
2743:
2708:
2675:
2646:
2613:
2574:
2548:
2506:
2473:
2432:
2397:
2374:
2354:
2289:
2168:
2107:
2075:
2055:
2035:
2015:
1963:
1911:
1859:
1839:
1819:
1799:
1370:
1338:
954:
884:
780:
747:
675:
636:
609:
558:
538:
518:
457:
437:
399:
369:
352:{\displaystyle D(a_{ii},R_{i})\subseteq \mathbb {C} }
305:
230:
207:
176:
132:
102:
76:
53:
35:. It was first published by the Soviet mathematician
3481:It would be good to reduce the condition number of
2009:â it is easy to see that the disc for row 2 covers
731:{\displaystyle \sum _{j}a_{ij}x_{j}=\lambda x_{i}.}
4073:
3984:
3946:
3908:
3789:
3705:
3393:
3363:
3331:
3302:
3269:
3228:
3195:
3159:
2984:
2955:
2919:
2871:
2832:
2793:
2764:
2723:
2690:
2661:
2628:
2589:
2560:
2530:
2488:
2447:
2418:
2380:
2360:
2337:
2221:
2154:
2081:
2061:
2041:
2021:
2001:
1949:
1897:
1845:
1825:
1805:
1782:
1353:
1256:
933:
863:
763:
730:
654:
622:
583:
544:
524:
498:
443:
412:
385:
351:
288:
213:
189:
162:
118:
88:
59:
4195:"Ăber die Abgrenzung der Eigenwerte einer Matrix"
451:lies within at least one of the Gershgorin discs
4311:
4146:Roger A. Horn & Charles R. Johnson (2013),
4074:{\textstyle |a_{ii}|>\sum _{j\neq i}|a_{ji}|}
4300:." From MathWorld—A Wolfram Web Resource.
2096:
2239:discs is disjoint from the union of the other
4249:
4175:Eigenvalue continuity and Gersgorin's theorem
3790:{\displaystyle \sum _{j\neq i}|a_{ij}|=R_{i}}
2247:discs then the former union contains exactly
4090:For matrices with non-negative entries, see
2496:is disjoint from the union of the remaining
157:
139:
4189:
3474:is 1000 then we can only be confident that
420:. Such a disc is called a Gershgorin disc.
4179:Electronic Journal of Linear Algebra (ELA)
3881:
3875:
3853:
3831:
1275:must also lie within the Gershgorin discs
4303:Semyon Aranovich Gershgorin biography at
4199:Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk
3381:
3216:
1341:
345:
4173:Chi-Kwong Li & Fuzhen Zhang (2019),
3565:
1793:â which by construction has eigenvalues
666:th component of that equation to get:
4312:
3501:is constructed, and then the equation
42:
4236:
4210:
3310:as a composition of roots solver and
3203:should be understood in the sense of
3800:to obtain the following four discs:
201:of the non-diagonal entries in the
163:{\displaystyle i\in \{1,\dots ,n\}}
13:
2996:discs, and the theorem is proven.
2920:{\displaystyle 0<d(t_{0})<d}
14:
4341:
4276:
2180:
2119:
1969:
1917:
1865:
3394:{\displaystyle \mathbb {C} ^{n}}
3229:{\displaystyle \mathbb {C} ^{n}}
2049:while the disc for row 3 covers
1354:{\displaystyle \mathbb {C} ^{n}}
1284:corresponding to the columns of
499:{\displaystyle D(a_{ii},R_{i}).}
4246:. 1st ed., Prentice Hall, 1962.
4244:(2nd ed.), Springer-Verlag
2956:{\displaystyle \lambda (t_{0})}
2872:{\displaystyle 0<t_{0}<1}
2338:{\displaystyle B(t)=(1-t)D+tA.}
552:with corresponding eigenvector
4167:
4140:
4067:
4049:
4025:
4007:
3979:
3964:
3941:
3929:
3900:
3885:
3869:
3857:
3847:
3835:
3825:
3813:
3770:
3752:
3426:
3358:
3352:
3326:
3320:
3297:
3291:
3190:
3184:
2979:
2973:
2950:
2937:
2908:
2895:
2821:
2815:
2788:
2782:
2753:
2747:
2718:
2712:
2685:
2679:
2656:
2650:
2623:
2617:
2584:
2578:
2525:
2513:
2483:
2477:
2442:
2436:
2413:
2407:
2317:
2305:
2299:
2293:
1769:
1757:
1746:
1734:
1713:
1701:
1693:
1681:
845:
823:
578:
565:
490:
461:
338:
309:
1:
4284:"Gershgorin's circle theorem"
4133:
3270:{\displaystyle a_{n}\equiv 1}
2636:are a continuous function of
1312:
3539:is the identity matrix, the
2731:from the union of the other
2597:is a decreasing function of
2388:, which is a contradiction.
2097:Strengthening of the theorem
662:, in particular we take the
655:{\displaystyle Ax=\lambda x}
7:
4218:, Berlin: Springer-Verlag,
4084:
3364:{\displaystyle \lambda (t)}
3303:{\displaystyle \lambda (t)}
3196:{\displaystyle \lambda (t)}
2985:{\displaystyle \lambda (1)}
2794:{\displaystyle \lambda (1)}
2662:{\displaystyle \lambda (t)}
2601:, so it is always at least
37:Semyon Aronovich Gershgorin
10:
4346:
4181:{Vol.35, pp.619-625|2019}
4152:Cambridge University Press
3997:diagonally dominant matrix
3985:{\displaystyle D(-11,2.2)}
3558:
3277:), which can be proved an
3242:characteristic polynomials
2765:{\displaystyle d(0)\geq d}
2426:. The diagonal entries of
2391:The statement is true for
4298:Gershgorin Circle Theorem
4242:Matrix Iterative Analysis
4216:GerĹĄgorin and His Circles
3451:is a matrix with a large
2992:lies in the union of the
2801:lies in the union of the
2735:discs is also continuous.
2607:Since the eigenvalues of
595:such that the element of
584:{\displaystyle x=(x_{j})}
89:{\displaystyle n\times n}
27:may be used to bound the
25:Gershgorin circle theorem
4098:Doubly stochastic matrix
4092:PerronâFrobenius theorem
3947:{\displaystyle D(2,1.2)}
874:Therefore, applying the
525:{\displaystyle \lambda }
941:based on how we picked
4123:Bendixson's inequality
4075:
3986:
3948:
3910:
3791:
3707:
3572:
3485:. This can be done by
3395:
3365:
3342:Individual eigenvalue
3333:
3304:
3271:
3230:
3197:
3161:
2986:
2957:
2921:
2873:
2834:
2833:{\displaystyle d(1)=0}
2795:
2766:
2725:
2692:
2663:
2630:
2591:
2562:
2561:{\displaystyle d>0}
2532:
2490:
2449:
2420:
2419:{\displaystyle D=B(0)}
2382:
2362:
2339:
2223:
2156:
2083:
2063:
2043:
2023:
2003:
1951:
1899:
1847:
1827:
1807:
1784:
1355:
1258:
935:
865:
765:
764:{\displaystyle a_{ii}}
732:
656:
624:
585:
546:
526:
500:
445:
414:
387:
386:{\displaystyle a_{ii}}
353:
290:
215:
191:
164:
120:
119:{\displaystyle a_{ij}}
90:
61:
4118:Muirhead's inequality
4076:
3987:
3949:
3911:
3792:
3708:
3569:
3396:
3366:
3334:
3305:
3272:
3231:
3198:
3162:
3003:Consider the matrix,
2987:
2958:
2922:
2874:
2835:
2796:
2767:
2726:
2693:
2664:
2640:, for any eigenvalue
2631:
2592:
2563:
2533:
2531:{\displaystyle t\in }
2491:
2455:are equal to that of
2450:
2421:
2383:
2363:
2340:
2224:
2157:
2084:
2064:
2044:
2024:
2004:
1952:
1900:
1848:
1828:
1808:
1785:
1356:
1295:Apply the Theorem to
1259:
936:
866:
766:
733:
657:
625:
623:{\displaystyle x_{i}}
586:
547:
527:
501:
446:
415:
413:{\displaystyle R_{i}}
388:
354:
291:
216:
192:
190:{\displaystyle R_{i}}
165:
121:
96:matrix, with entries
91:
62:
4296:Eric W. Weisstein. "
4003:
3958:
3923:
3807:
3732:
3579:
3376:
3346:
3332:{\displaystyle B(t)}
3314:
3285:
3248:
3211:
3178:
3007:
2967:
2931:
2883:
2844:
2809:
2776:
2741:
2724:{\displaystyle d(t)}
2706:
2698:in the union of the
2691:{\displaystyle B(t)}
2673:
2644:
2629:{\displaystyle B(t)}
2611:
2590:{\displaystyle B(t)}
2572:
2546:
2504:
2489:{\displaystyle B(t)}
2471:
2448:{\displaystyle B(t)}
2430:
2395:
2372:
2352:
2287:
2166:
2105:
2073:
2053:
2033:
2013:
1961:
1909:
1857:
1837:
1817:
1797:
1368:
1336:
952:
882:
778:
745:
673:
634:
607:
556:
536:
532:be an eigenvalue of
516:
455:
435:
397:
367:
303:
228:
205:
174:
130:
100:
74:
51:
16:Bound on eigenvalues
4320:Theorems in algebra
4259:Matrix Computations
3466:. For instance, if
3339:is also continuous.
2702:discs its distance
2568:. The distance for
1271:The eigenvalues of
878:and recalling that
876:triangle inequality
771:to the other side:
43:Statement and proof
4150:, second edition,
4128:SchurâHorn theorem
4071:
4047:
3982:
3944:
3906:
3787:
3750:
3703:
3694:
3573:
3460:order of magnitude
3416:argument principle
3391:
3361:
3329:
3300:
3267:
3226:
3193:
3174:The continuity of
3157:
3151:
2982:
2953:
2917:
2869:
2840:, so there exists
2830:
2791:
2762:
2721:
2688:
2659:
2626:
2587:
2558:
2528:
2486:
2445:
2416:
2378:
2358:
2335:
2235:: If the union of
2219:
2213:
2212:
2152:
2146:
2145:
2079:
2059:
2039:
2019:
1999:
1993:
1992:
1947:
1941:
1940:
1895:
1889:
1888:
1853:with eigenvectors
1843:
1823:
1803:
1780:
1774:
1555:
1491:
1428:
1351:
1254:
1216:
1136:
1070:
1007:
931:
861:
796:
761:
728:
685:
652:
620:
581:
542:
522:
496:
441:
410:
383:
349:
286:
261:
211:
197:be the sum of the
187:
160:
116:
86:
57:
4238:Varga, Richard S.
4212:Varga, Richard S.
4032:
3879:
3735:
2927:. But this means
2381:{\displaystyle t}
2361:{\displaystyle t}
2082:{\displaystyle c}
2062:{\displaystyle a}
2042:{\displaystyle b}
2022:{\displaystyle a}
1846:{\displaystyle c}
1826:{\displaystyle b}
1806:{\displaystyle a}
1324:The theorem does
1201:
1196:
1121:
1112:
1055:
1045:
992:
923:
781:
676:
599:with the largest
545:{\displaystyle A}
444:{\displaystyle A}
244:
214:{\displaystyle i}
60:{\displaystyle A}
4337:
4293:
4271:
4245:
4228:
4206:
4182:
4171:
4165:
4144:
4108:Joel Lee Brenner
4080:
4078:
4077:
4072:
4070:
4065:
4064:
4052:
4046:
4028:
4023:
4022:
4010:
3991:
3989:
3988:
3983:
3953:
3951:
3950:
3945:
3915:
3913:
3912:
3907:
3880:
3877:
3796:
3794:
3793:
3788:
3786:
3785:
3773:
3768:
3767:
3755:
3749:
3712:
3710:
3709:
3704:
3699:
3698:
3453:condition number
3447:is a vector and
3420:complex analysis
3400:
3398:
3397:
3392:
3390:
3389:
3384:
3370:
3368:
3367:
3362:
3338:
3336:
3335:
3330:
3309:
3307:
3306:
3301:
3276:
3274:
3273:
3268:
3260:
3259:
3238:Vieta's formulas
3235:
3233:
3232:
3227:
3225:
3224:
3219:
3202:
3200:
3199:
3194:
3166:
3164:
3163:
3158:
3156:
3155:
2991:
2989:
2988:
2983:
2962:
2960:
2959:
2954:
2949:
2948:
2926:
2924:
2923:
2918:
2907:
2906:
2878:
2876:
2875:
2870:
2862:
2861:
2839:
2837:
2836:
2831:
2800:
2798:
2797:
2792:
2771:
2769:
2768:
2763:
2730:
2728:
2727:
2722:
2697:
2695:
2694:
2689:
2668:
2666:
2665:
2660:
2635:
2633:
2632:
2627:
2596:
2594:
2593:
2588:
2567:
2565:
2564:
2559:
2537:
2535:
2534:
2529:
2495:
2493:
2492:
2487:
2454:
2452:
2451:
2446:
2425:
2423:
2422:
2417:
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2344:
2342:
2341:
2336:
2228:
2226:
2225:
2220:
2218:
2214:
2161:
2159:
2158:
2153:
2151:
2147:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1998:
1994:
1956:
1954:
1953:
1948:
1946:
1942:
1904:
1902:
1901:
1896:
1894:
1890:
1852:
1850:
1849:
1844:
1832:
1830:
1829:
1824:
1812:
1810:
1809:
1804:
1789:
1787:
1786:
1781:
1779:
1778:
1569:
1568:
1560:
1559:
1496:
1495:
1433:
1432:
1360:
1358:
1357:
1352:
1350:
1349:
1344:
1263:
1261:
1260:
1255:
1250:
1249:
1237:
1233:
1232:
1215:
1197:
1195:
1191:
1190:
1177:
1173:
1172:
1159:
1157:
1153:
1152:
1135:
1117:
1113:
1111:
1110:
1101:
1100:
1099:
1090:
1089:
1076:
1069:
1051:
1047:
1046:
1044:
1043:
1034:
1033:
1032:
1023:
1022:
1009:
1006:
983:
979:
978:
977:
940:
938:
937:
932:
924:
922:
918:
917:
904:
900:
899:
886:
870:
868:
867:
862:
857:
856:
844:
843:
819:
818:
809:
808:
795:
770:
768:
767:
762:
760:
759:
737:
735:
734:
729:
724:
723:
708:
707:
698:
697:
684:
661:
659:
658:
653:
629:
627:
626:
621:
619:
618:
590:
588:
587:
582:
577:
576:
551:
549:
548:
543:
531:
529:
528:
523:
505:
503:
502:
497:
489:
488:
476:
475:
450:
448:
447:
442:
419:
417:
416:
411:
409:
408:
392:
390:
389:
384:
382:
381:
358:
356:
355:
350:
348:
337:
336:
324:
323:
295:
293:
292:
287:
282:
278:
277:
260:
259:
240:
239:
220:
218:
217:
212:
196:
194:
193:
188:
186:
185:
169:
167:
166:
161:
125:
123:
122:
117:
115:
114:
95:
93:
92:
87:
66:
64:
63:
58:
4345:
4344:
4340:
4339:
4338:
4336:
4335:
4334:
4310:
4309:
4282:
4279:
4269:
4255:Van Loan, C. F.
4226:
4191:Gerschgorin, S.
4186:
4185:
4172:
4168:
4148:Matrix Analysis
4145:
4141:
4136:
4087:
4066:
4057:
4053:
4048:
4036:
4024:
4015:
4011:
4006:
4004:
4001:
4000:
3959:
3956:
3955:
3924:
3921:
3920:
3876:
3808:
3805:
3804:
3781:
3777:
3769:
3760:
3756:
3751:
3739:
3733:
3730:
3729:
3724:
3693:
3692:
3684:
3676:
3668:
3659:
3658:
3653:
3648:
3643:
3637:
3636:
3631:
3626:
3621:
3615:
3614:
3609:
3604:
3599:
3589:
3588:
3580:
3577:
3576:
3561:
3487:preconditioning
3429:
3385:
3380:
3379:
3377:
3374:
3373:
3347:
3344:
3343:
3315:
3312:
3311:
3286:
3283:
3282:
3255:
3251:
3249:
3246:
3245:
3220:
3215:
3214:
3212:
3209:
3208:
3179:
3176:
3175:
3150:
3149:
3144:
3139:
3134:
3129:
3123:
3122:
3117:
3112:
3107:
3102:
3096:
3095:
3090:
3085:
3080:
3075:
3069:
3068:
3063:
3058:
3053:
3048:
3042:
3041:
3036:
3031:
3026:
3021:
3011:
3010:
3008:
3005:
3004:
2999:
2968:
2965:
2964:
2944:
2940:
2932:
2929:
2928:
2902:
2898:
2884:
2881:
2880:
2857:
2853:
2845:
2842:
2841:
2810:
2807:
2806:
2777:
2774:
2773:
2742:
2739:
2738:
2707:
2704:
2703:
2674:
2671:
2670:
2645:
2642:
2641:
2612:
2609:
2608:
2573:
2570:
2569:
2547:
2544:
2543:
2505:
2502:
2501:
2472:
2469:
2468:
2431:
2428:
2427:
2396:
2393:
2392:
2373:
2370:
2369:
2353:
2350:
2349:
2288:
2285:
2284:
2259:eigenvalues of
2251:and the latter
2211:
2210:
2202:
2196:
2195:
2187:
2179:
2175:
2167:
2164:
2163:
2144:
2143:
2138:
2132:
2131:
2126:
2118:
2114:
2106:
2103:
2102:
2099:
2074:
2071:
2070:
2054:
2051:
2050:
2034:
2031:
2030:
2014:
2011:
2010:
1991:
1990:
1984:
1983:
1977:
1976:
1968:
1964:
1962:
1959:
1958:
1939:
1938:
1932:
1931:
1925:
1924:
1916:
1912:
1910:
1907:
1906:
1887:
1886:
1880:
1879:
1873:
1872:
1864:
1860:
1858:
1855:
1854:
1838:
1835:
1834:
1818:
1815:
1814:
1798:
1795:
1794:
1773:
1772:
1749:
1729:
1717:
1716:
1696:
1673:
1661:
1660:
1634:
1608:
1574:
1573:
1561:
1554:
1553:
1548:
1543:
1537:
1536:
1531:
1526:
1520:
1519:
1514:
1509:
1499:
1498:
1497:
1490:
1489:
1484:
1479:
1473:
1472:
1467:
1462:
1456:
1455:
1450:
1445:
1435:
1434:
1427:
1426:
1421:
1416:
1410:
1409:
1404:
1399:
1393:
1392:
1387:
1382:
1372:
1371:
1369:
1366:
1365:
1345:
1340:
1339:
1337:
1334:
1333:
1315:
1307:diagonal matrix
1283:
1245:
1241:
1225:
1221:
1217:
1205:
1186:
1182:
1178:
1168:
1164:
1160:
1158:
1145:
1141:
1137:
1125:
1106:
1102:
1095:
1091:
1082:
1078:
1077:
1075:
1071:
1059:
1039:
1035:
1028:
1024:
1015:
1011:
1010:
1008:
996:
991:
987:
970:
966:
959:
955:
953:
950:
949:
913:
909:
905:
895:
891:
887:
885:
883:
880:
879:
852:
848:
836:
832:
814:
810:
801:
797:
785:
779:
776:
775:
752:
748:
746:
743:
742:
719:
715:
703:
699:
690:
686:
680:
674:
671:
670:
635:
632:
631:
614:
610:
608:
605:
604:
572:
568:
557:
554:
553:
537:
534:
533:
517:
514:
513:
484:
480:
468:
464:
456:
453:
452:
436:
433:
432:
404:
400:
398:
395:
394:
374:
370:
368:
365:
364:
344:
332:
328:
316:
312:
304:
301:
300:
270:
266:
262:
255:
248:
235:
231:
229:
226:
225:
206:
203:
202:
199:absolute values
181:
177:
175:
172:
171:
131:
128:
127:
107:
103:
101:
98:
97:
75:
72:
71:
52:
49:
48:
45:
17:
12:
11:
5:
4343:
4333:
4332:
4327:
4325:Linear algebra
4322:
4308:
4307:
4301:
4294:
4278:
4277:External links
4275:
4274:
4273:
4267:
4247:
4234:
4224:
4208:
4184:
4183:
4166:
4138:
4137:
4135:
4132:
4131:
4130:
4125:
4120:
4115:
4113:Metzler matrix
4110:
4105:
4103:Hurwitz matrix
4100:
4095:
4086:
4083:
4069:
4063:
4060:
4056:
4051:
4045:
4042:
4039:
4035:
4031:
4027:
4021:
4018:
4014:
4009:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3943:
3940:
3937:
3934:
3931:
3928:
3917:
3916:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3798:
3797:
3784:
3780:
3776:
3772:
3766:
3763:
3759:
3754:
3748:
3745:
3742:
3738:
3720:
3714:
3713:
3702:
3697:
3691:
3688:
3685:
3683:
3680:
3677:
3675:
3672:
3669:
3667:
3664:
3661:
3660:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3638:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3594:
3592:
3587:
3584:
3560:
3557:
3509:is solved for
3428:
3425:
3424:
3423:
3405:Added Remark:
3403:
3402:
3388:
3383:
3360:
3357:
3354:
3351:
3340:
3328:
3325:
3322:
3319:
3299:
3296:
3293:
3290:
3266:
3263:
3258:
3254:
3223:
3218:
3192:
3189:
3186:
3183:
3154:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3124:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3097:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3070:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3043:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3016:
3014:
2981:
2978:
2975:
2972:
2952:
2947:
2943:
2939:
2936:
2916:
2913:
2910:
2905:
2901:
2897:
2894:
2891:
2888:
2868:
2865:
2860:
2856:
2852:
2849:
2829:
2826:
2823:
2820:
2817:
2814:
2790:
2787:
2784:
2781:
2761:
2758:
2755:
2752:
2749:
2746:
2720:
2717:
2714:
2711:
2687:
2684:
2681:
2678:
2658:
2655:
2652:
2649:
2625:
2622:
2619:
2616:
2586:
2583:
2580:
2577:
2557:
2554:
2551:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2485:
2482:
2479:
2476:
2444:
2441:
2438:
2435:
2415:
2412:
2409:
2406:
2403:
2400:
2377:
2357:
2346:
2345:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2266:
2217:
2209:
2206:
2203:
2201:
2198:
2197:
2194:
2191:
2188:
2186:
2183:
2182:
2178:
2174:
2171:
2150:
2142:
2139:
2137:
2134:
2133:
2130:
2127:
2125:
2122:
2121:
2117:
2113:
2110:
2098:
2095:
2078:
2058:
2038:
2018:
1997:
1989:
1986:
1985:
1982:
1979:
1978:
1975:
1972:
1971:
1967:
1945:
1937:
1934:
1933:
1930:
1927:
1926:
1923:
1920:
1919:
1915:
1893:
1885:
1882:
1881:
1878:
1875:
1874:
1871:
1868:
1867:
1863:
1842:
1822:
1802:
1791:
1790:
1777:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1728:
1725:
1722:
1719:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1672:
1669:
1666:
1663:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1579:
1577:
1572:
1567:
1564:
1558:
1552:
1549:
1547:
1544:
1542:
1539:
1538:
1535:
1532:
1530:
1527:
1525:
1522:
1521:
1518:
1515:
1513:
1510:
1508:
1505:
1504:
1502:
1494:
1488:
1485:
1483:
1480:
1478:
1475:
1474:
1471:
1468:
1466:
1463:
1461:
1458:
1457:
1454:
1451:
1449:
1446:
1444:
1441:
1440:
1438:
1431:
1425:
1422:
1420:
1417:
1415:
1412:
1411:
1408:
1405:
1403:
1400:
1398:
1395:
1394:
1391:
1388:
1386:
1383:
1381:
1378:
1377:
1375:
1348:
1343:
1314:
1311:
1290:
1289:
1279:
1265:
1264:
1253:
1248:
1244:
1240:
1236:
1231:
1228:
1224:
1220:
1214:
1211:
1208:
1204:
1200:
1194:
1189:
1185:
1181:
1176:
1171:
1167:
1163:
1156:
1151:
1148:
1144:
1140:
1134:
1131:
1128:
1124:
1120:
1116:
1109:
1105:
1098:
1094:
1088:
1085:
1081:
1074:
1068:
1065:
1062:
1058:
1054:
1050:
1042:
1038:
1031:
1027:
1021:
1018:
1014:
1005:
1002:
999:
995:
990:
986:
982:
976:
973:
969:
965:
962:
958:
930:
927:
921:
916:
912:
908:
903:
898:
894:
890:
872:
871:
860:
855:
851:
847:
842:
839:
835:
831:
828:
825:
822:
817:
813:
807:
804:
800:
794:
791:
788:
784:
758:
755:
751:
739:
738:
727:
722:
718:
714:
711:
706:
702:
696:
693:
689:
683:
679:
651:
648:
645:
642:
639:
617:
613:
601:absolute value
580:
575:
571:
567:
564:
561:
541:
521:
507:
506:
495:
492:
487:
483:
479:
474:
471:
467:
463:
460:
440:
407:
403:
380:
377:
373:
347:
343:
340:
335:
331:
327:
322:
319:
315:
311:
308:
297:
296:
285:
281:
276:
273:
269:
265:
258:
254:
251:
247:
243:
238:
234:
210:
184:
180:
159:
156:
153:
150:
147:
144:
141:
138:
135:
113:
110:
106:
85:
82:
79:
56:
44:
41:
15:
9:
6:
4:
3:
2:
4342:
4331:
4330:Matrix theory
4328:
4326:
4323:
4321:
4318:
4317:
4315:
4306:
4302:
4299:
4295:
4291:
4290:
4285:
4281:
4280:
4270:
4268:0-8018-5413-X
4264:
4260:
4256:
4252:
4248:
4243:
4239:
4235:
4232:
4227:
4225:3-540-21100-4
4221:
4217:
4213:
4209:
4204:
4201:(in German),
4200:
4196:
4192:
4188:
4187:
4180:
4176:
4170:
4164:
4160:
4159:9780521548236
4156:
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4149:
4143:
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4126:
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4119:
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4114:
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4089:
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4082:
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4040:
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4029:
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3993:
3976:
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3926:
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3897:
3894:
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3803:
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3778:
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3727:
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3645:
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3141:
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2858:
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2827:
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2812:
2804:
2785:
2779:
2772:, and assume
2759:
2756:
2750:
2744:
2736:
2732:
2715:
2709:
2699:
2682:
2676:
2653:
2647:
2637:
2620:
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2410:
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2389:
2375:
2355:
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2329:
2326:
2323:
2320:
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2308:
2302:
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2254:
2250:
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2242:
2238:
2234:
2230:
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2199:
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2172:
2169:
2148:
2140:
2135:
2128:
2123:
2115:
2111:
2108:
2094:
2092:
2076:
2056:
2036:
2016:
1995:
1987:
1980:
1973:
1965:
1943:
1935:
1928:
1921:
1913:
1891:
1883:
1876:
1869:
1861:
1840:
1820:
1800:
1775:
1766:
1763:
1760:
1754:
1751:
1743:
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1737:
1731:
1726:
1723:
1720:
1710:
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1690:
1687:
1684:
1678:
1675:
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1613:
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1599:
1596:
1593:
1590:
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1584:
1581:
1575:
1570:
1565:
1562:
1556:
1550:
1545:
1540:
1533:
1528:
1523:
1516:
1511:
1506:
1500:
1492:
1486:
1481:
1476:
1469:
1464:
1459:
1452:
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1436:
1429:
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1327:
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1298:
1294:
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1282:
1278:
1274:
1270:
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1266:
1251:
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1238:
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1229:
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1222:
1218:
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1019:
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756:
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749:
725:
720:
716:
712:
709:
704:
700:
694:
691:
687:
681:
677:
669:
668:
667:
665:
649:
646:
643:
640:
637:
615:
611:
602:
598:
594:
573:
569:
562:
559:
539:
519:
511:
493:
485:
481:
477:
472:
469:
465:
458:
438:
430:
426:
423:
422:
421:
405:
401:
378:
375:
371:
362:
341:
333:
329:
325:
320:
317:
313:
306:
283:
279:
274:
271:
267:
263:
256:
252:
249:
245:
241:
236:
232:
224:
223:
222:
208:
200:
182:
178:
154:
151:
148:
145:
142:
136:
133:
111:
108:
104:
83:
80:
77:
70:
54:
40:
38:
34:
30:
26:
22:
4287:
4258:
4251:Golub, G. H.
4241:
4215:
4202:
4198:
4174:
4169:
4147:
4142:
3994:
3918:
3799:
3721:
3717:
3715:
3562:
3552:
3548:
3544:
3536:
3532:
3528:
3526:
3521:
3514:
3513:. Using the
3510:
3506:
3502:
3498:
3494:
3490:
3482:
3480:
3475:
3471:
3467:
3463:
3457:
3448:
3444:
3440:
3436:
3432:
3430:
3411:
3404:
3168:
3002:
2998:
2993:
2805:discs. Then
2802:
2734:
2701:
2639:
2606:
2602:
2598:
2539:
2497:
2464:
2460:
2456:
2390:
2347:
2277:
2273:
2269:
2268:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2231:
2100:
2090:
1792:
1329:
1325:
1323:
1316:
1302:
1301:
1296:
1292:
1291:
1285:
1280:
1276:
1272:
1268:
942:
873:
740:
663:
596:
592:
509:
508:
424:
393:with radius
363:centered at
359:be a closed
298:
46:
31:of a square
24:
18:
3541:eigenvalues
3527:Now, since
3489:: A matrix
3427:Application
221:-th row:
21:mathematics
4314:Categories
4289:PlanetMath
4134:References
3493:such that
3240:(note for
2879:such that
2737:Obviously
1313:Discussion
1269:Corollary.
429:eigenvalue
4205:: 749â754
4041:≠
4034:∑
3968:−
3889:−
3744:≠
3737:∑
3687:−
3679:−
3671:−
3663:−
3350:λ
3289:λ
3262:≡
3182:λ
2971:λ
2935:λ
2780:λ
2757:≥
2648:λ
2511:∈
2467:discs of
2312:−
2205:−
2190:−
1764:−
1741:−
1724:−
1708:−
1688:−
1668:−
1652:−
1643:−
1626:−
1617:−
1582:−
1563:−
1210:≠
1203:∑
1199:≤
1130:≠
1123:∑
1064:≠
1057:∑
1053:≤
1001:≠
994:∑
964:−
961:λ
926:≤
830:−
827:λ
790:≠
783:∑
713:λ
678:∑
647:λ
520:λ
342:⊆
253:≠
246:∑
149:…
137:∈
81:×
4305:MacTutor
4257:(1996),
4240:(2002),
4214:(2004),
4193:(1931),
4085:See also
3279:open map
3205:topology
2500:for all
2280:and let
1303:Example.
630:. Since
425:Theorem.
29:spectrum
3559:Example
3518:inverse
2233:Theorem
741:Taking
591:. Find
69:complex
4265:
4231:Errata
4222:
4157:
3535:where
3443:where
2272:: Let
1957:, and
1833:, and
1305:For a
1293:Proof.
510:Proof.
427:Every
126:. For
33:matrix
23:, the
3571:each.
3555:was.
3515:exact
3244:that
2270:Proof
1319:norms
67:be a
4263:ISBN
4220:ISBN
4155:ISBN
4030:>
3954:and
3439:for
2912:<
2890:<
2864:<
2851:<
2553:>
2069:and
2029:and
1330:axes
512:Let
361:disc
299:Let
170:let
47:Let
4229:. (
3977:2.2
3939:1.2
3878:and
3845:0.6
3634:0.2
3629:0.2
3619:0.2
3543:of
3520:of
3503:PAx
3418:of
2803:n-k
2733:n-k
2669:of
2542:is
2498:n-k
2162:or
2091:sum
1332:in
1326:not
603:is
431:of
19:In
4316::
4286:.
4253:;
4233:).
4197:,
4177:,
3999::
3992:.
3971:11
3892:11
3817:10
3722:ii
3690:11
3597:10
3549:PA
3545:PA
3531:â
3529:PA
3507:Pb
3505:=
3497:â
3455:.
3435:=
3433:Ax
2605:.
2263:,
1905:,
1813:,
945:,
4292:.
4272:.
4207:.
4203:6
4161:[
4094:.
4068:|
4062:i
4059:j
4055:a
4050:|
4044:i
4038:j
4026:|
4020:i
4017:i
4013:a
4008:|
3980:)
3974:,
3965:(
3962:D
3942:)
3936:,
3933:2
3930:(
3927:D
3904:.
3901:)
3898:3
3895:,
3886:(
3883:D
3873:,
3870:)
3867:3
3864:,
3861:2
3858:(
3855:D
3851:,
3848:)
3842:,
3839:8
3836:(
3833:D
3829:,
3826:)
3823:2
3820:,
3814:(
3811:D
3783:i
3779:R
3775:=
3771:|
3765:j
3762:i
3758:a
3753:|
3747:i
3741:j
3718:a
3701:.
3696:]
3682:1
3674:1
3666:1
3656:1
3651:2
3646:1
3641:1
3624:8
3612:1
3607:0
3602:1
3591:[
3586:=
3583:A
3553:P
3537:I
3533:I
3522:A
3511:x
3499:A
3495:P
3491:P
3483:A
3476:x
3472:A
3468:b
3464:A
3449:A
3445:b
3441:x
3437:b
3412:C
3387:n
3382:C
3359:)
3356:t
3353:(
3327:)
3324:t
3321:(
3318:B
3298:)
3295:t
3292:(
3265:1
3257:n
3253:a
3222:n
3217:C
3191:)
3188:t
3185:(
3153:]
3147:1
3142:0
3137:0
3132:0
3127:0
3120:1
3115:1
3110:0
3105:0
3100:0
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3078:0
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3066:0
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3024:1
3019:5
3013:[
2994:k
2980:)
2977:1
2974:(
2951:)
2946:0
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2938:(
2915:d
2909:)
2904:0
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2893:d
2887:0
2867:1
2859:0
2855:t
2848:0
2828:0
2825:=
2822:)
2819:1
2816:(
2813:d
2789:)
2786:1
2783:(
2760:d
2754:)
2751:0
2748:(
2745:d
2719:)
2716:t
2713:(
2710:d
2700:k
2686:)
2683:t
2680:(
2677:B
2657:)
2654:t
2651:(
2638:t
2624:)
2621:t
2618:(
2615:B
2603:d
2599:t
2585:)
2582:t
2579:(
2576:B
2556:0
2550:d
2540:A
2526:]
2523:1
2520:,
2517:0
2514:[
2508:t
2484:)
2481:t
2478:(
2475:B
2465:k
2461:t
2457:A
2443:)
2440:t
2437:(
2434:B
2414:)
2411:0
2408:(
2405:B
2402:=
2399:D
2376:t
2356:t
2333:.
2330:A
2327:t
2324:+
2321:D
2318:)
2315:t
2309:1
2306:(
2303:=
2300:)
2297:t
2294:(
2291:B
2278:A
2274:D
2261:A
2257:k
2253:n
2249:k
2245:k
2241:n
2237:k
2216:)
2208:1
2200:1
2193:2
2185:1
2177:(
2173:=
2170:A
2149:)
2141:0
2136:4
2129:1
2124:0
2116:(
2112:=
2109:A
2077:c
2057:a
2037:b
2017:a
1996:)
1988:1
1981:0
1974:2
1966:(
1944:)
1936:0
1929:1
1922:2
1914:(
1892:)
1884:1
1877:1
1870:3
1862:(
1841:c
1821:b
1801:a
1776:)
1770:)
1767:c
1761:a
1758:(
1755:+
1752:a
1747:)
1744:c
1738:a
1735:(
1732:2
1727:a
1721:c
1714:)
1711:b
1705:a
1702:(
1699:2
1694:)
1691:b
1685:a
1682:(
1679:+
1676:a
1671:a
1665:b
1658:c
1655:2
1649:b
1646:4
1640:a
1637:6
1632:c
1629:4
1623:b
1620:2
1614:a
1611:6
1606:c
1603:2
1600:+
1597:b
1594:2
1591:+
1588:a
1585:3
1576:(
1571:=
1566:1
1557:)
1551:1
1546:0
1541:1
1534:0
1529:1
1524:1
1517:2
1512:2
1507:3
1501:(
1493:)
1487:c
1482:0
1477:0
1470:0
1465:b
1460:0
1453:0
1448:0
1443:a
1437:(
1430:)
1424:1
1419:0
1414:1
1407:0
1402:1
1397:1
1390:2
1385:2
1380:3
1374:(
1347:n
1342:C
1297:A
1288:.
1286:A
1281:j
1277:C
1273:A
1252:.
1247:i
1243:R
1239:=
1235:|
1230:j
1227:i
1223:a
1219:|
1213:i
1207:j
1193:|
1188:i
1184:x
1180:|
1175:|
1170:j
1166:x
1162:|
1155:|
1150:j
1147:i
1143:a
1139:|
1133:i
1127:j
1119:=
1115:|
1108:i
1104:x
1097:j
1093:x
1087:j
1084:i
1080:a
1073:|
1067:i
1061:j
1049:|
1041:i
1037:x
1030:j
1026:x
1020:j
1017:i
1013:a
1004:i
998:j
989:|
985:=
981:|
975:i
972:i
968:a
957:|
943:i
929:1
920:|
915:i
911:x
907:|
902:|
897:j
893:x
889:|
859:.
854:i
850:x
846:)
841:i
838:i
834:a
824:(
821:=
816:j
812:x
806:j
803:i
799:a
793:i
787:j
757:i
754:i
750:a
726:.
721:i
717:x
710:=
705:j
701:x
695:j
692:i
688:a
682:j
664:i
650:x
644:=
641:x
638:A
616:i
612:x
597:x
593:i
579:)
574:j
570:x
566:(
563:=
560:x
540:A
494:.
491:)
486:i
482:R
478:,
473:i
470:i
466:a
462:(
459:D
439:A
406:i
402:R
379:i
376:i
372:a
346:C
339:)
334:i
330:R
326:,
321:i
318:i
314:a
310:(
307:D
284:.
280:|
275:j
272:i
268:a
264:|
257:i
250:j
242:=
237:i
233:R
209:i
183:i
179:R
158:}
155:n
152:,
146:,
143:1
140:{
134:i
112:j
109:i
105:a
84:n
78:n
55:A
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