AU = U λ c 2 c 3 c n C C n,, n U 2 U2 C U 2 = B = b 22 b 23 b 2n b 33 b 3n b nn U = U ( U 2, U AU = = = ( ( U 2 U 2 U AU ( U2 λ λ d 2 d 3 d n b 22 b 2

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1 Jordan, A m? (264(, A A m, A (, P P AP = D, A m = P D m P, P AP 837, Jacobi (, ( Jacobi,, Schur 24 Cayley-Hamilton 25,, A m Schur Jordan 26 Schur : 3 (Schur ( A C n n, U U AU = B, (3 B A n n =, n, n λ C A, α,, α C n α, α 2,, α n, U = (α, α 2,, α n Aα = λα, Aα 2,, Aα n α, α 2,, α n, λ c 2 c 3 c n c 22 c 23 c 2n AU = A(α, α 2,, α n = (α, α 2,, α n, c n2 c n3 c nn c ij c 22 c 23 c 2n C = c n2 c n3 c nn 24 Issai Schur (875-94,, Frobenius 25 Arthur Cayley(82-895,, William Hamilton(85-865,,, Hamilton, ( Quaternions 26 Marie Ennemond Camille Jordan ( ,, Jordan Curve Theorem camillejordan 74

2 AU = U λ c 2 c 3 c n C C n,, n U 2 U2 C U 2 = B = b 22 b 23 b 2n b 33 b 3n b nn U = U ( U 2, U AU = = = ( ( U 2 U 2 U AU ( U2 λ λ d 2 d 3 d n b 22 b 23 b 2n b 33 b 3n b nn c 2 c 3 c n C ( U 2 (3 A = UBU (32 (32 A Schur Schur (,,, Schur, P AP, P AP, ( F 3 A = (a ij F, p < q n P = I + αe pq, α F, B = (b ij = P AP A (, b pq = a pq + α(a pp a qq, a pp a qq, α b pq = 75

3 P, P = I αe pq P A A q α p, A ; AP B A ( b pq 3 λ λ 2, P = I c (λ λ 2 E 2, A = λ a b λ c λ 2, B = P AP = λ a x λ λ 2 Q = I x (λ λ 2 E 3, Q BQ B x A λ a λ λ 2 3, 3 32 ( Schur n A λi A = s (λ λ i n i s, σ(a = {λ i i s}, n i = n P i= P AP = A A 2 A s, A i λ i n i i=, Schur P? 2 Schur Cayley-Hamilton, 32 A n, A n = 33 (Cayley-Hamilton 27 A f(λ, f(a = A f(λ = (λ λ n (λ λ 2 n2 (λ λ s ns, 27 n = 2, 3 Cayley 853, n = 4 Hamilton 85, Frobenius

4 s i= n i = n, A B, g(x, g(a = g(b = Schur, A = A A 2 A s, A i λ i n i f(a = (A λ I n (A λ 2 I n2 (B λ s I n s 32, i, (A i λ i I ni n i =, i (A λ i I n i i n i, n A n, Cayley-Hamilton, A n,, A n n, m, A m Span{I, A, A 2,, A n }, n A I, A, A 2,, A n M n (C 33 A 2, A 3, A 4 A = ( 2 3 2, A f(λ = λ 2 4λ +, A 2 4A + I = A 2 = 4A I, A 3 = 4A 2 A = 5A 4I, A 4 = 5A 2 4A = 56A 5I 3 (Sylvester A B m n n m, m n λi m AB = λ m n λi n BA : ( I B I ( ( BA BAB = A AB A AB = ( BA A ( I B I,, C = ( BA A ( C 2 = A AB C BA m ; C 2 AB n C C 2, AB BA, m n 34 2 A = 3 4, B = (

5 2 AB = AB (, ( 2 2 BA =, 2 2 BA, -4, Sylvester AB,, -4 ( r(ab = 2, AB, BA 35 u n, n Householder I 2uu T Sylvester, λi (I 2uu T = (λ I + 2uu T = (λ n λ + 2u T u = (λ n (λ +, λ = n, λ =, tr(i 2uu T = n 2; I 2uu T = 36 A n, A, n, λi A = λ n (λ tr A Sylvester Cayley-Hamilton 34, m n m n, Sylvester, Cayley-Hamilton n n,, Cayley-Hamilton, ( d g(x g(a =, A A d 37 3 A f(λ = λ 3 5λ 2 g(x = x 2 5x g(a =, A 4 A 3 5A 2 =, A 4 = 25A 2 g(a = A 2 5A =, A 4 = 25A 3 A n, f(x f(a =, f(x A A, m A (x m(x, 5 A, A, (, 34 AB? 78

6 32 m(x A, f(x A, m(x f(x, m(x xi A f(x = m(xq(x + r(x, r(x m(x r(x = f(a = m(aq(a + r(a = r(a, r(x A m(x A, r(x =, m(x f(x 38 n : J n = A 2222 σ σ n, A n =, A n (, A m(x = x n, A J n n n n Jordan J n n n (, Jordan, J n ( 39 : ( a A = (a a a a a A f(λ = (λ a 6 32 m(x = (x a k, k 6, (A ai ( A ai = ( A ai,, m(x = (x a 3 3 a b, : A = ( a a a a a (b ( b b A (λ a 5 (λ b 3 A (λ a s (λ b t A ai A bi A ai = J 2 J 3 B, A bi = C ( J 2, 79

7 B C ( a b, A ai A bi 3 2, A m(x = (x a 3 (x b 2 33 A, m(x A λ C λ A m(λ =,,, λ A, α, = m(aα = m(λ α α, m(λ = 3 A A, A m = A x k, k m 32 : A = ( 2 2 ( 2 4 A f(λ = (λ 3(λ 2 3, : (x 3(x 2; (x 3(x 2 2 ; (x 3(x 2 3, (A 3I(A 2I ; (A 3I(A 2I 2 =, m(x = (x 3(x 2 2 ( 6: 34 ( : 35 A B, P B = P AP A B m A (x m B (x, = m B (B = m B (P AP = P m B (AP, m B (A =, m B (x A, m A (x m B (x;, m B (x m A (x, m A (x = m B (x 34 A A 34 35,, A m(x = 8

8 (x λ (x λ 2 (x λ s Schur, Schur A = A A 2 A s, A i λ i n i, s n i = n A i ( λ i I s s =, m(x = x λ, m(a = A = λ I, < s 34, B = A A 2 A s m(x, B m(a = m(b m(a s =, m(a s = A s, A s, A i= 3 A, f(x f(a =, A 33 A A 2 = A, m(x (x 2 x, ; A A 2 = I, m(x (x 2,, V C (, σ EndV, λ C α V σ(α = λα, λ σ, α σ ( λ λ σ, V {x V σ(x = λx} σ ( λ, V λ V, V λ = Ker(λI σ (33,,,, V σ, σ ( 7 35 V, σ EndV, A σ ( A σ (; (2 σ λ,, λ s, A V = s V λi A σ, σ = s σ i, σ i σ V λi i= ( θ, π ( cos θ sin θ A = sin θ cos θ 8 i=

9 ,, A,,, ±,, Schur,? 2, Schur? 3 A, B, AB BA,? 4? 5 σ, σ? 6 σ, σ? Jordan : Schur ( 32, N A = λi + N P AP = λi + P NP, 32 A n r(a = n, A J n ( B A A = T, B n, r(b = ( T r(a = n B, P = B, P AP = J n 322 A = C = J 2 J 2 ( J3 T 4, A B = ( J3 T, 2 r(a n 2 n A, B C 2 B C, B 2 C 2 = J n ( 6 32 ( J T n J n = ( I n, J n J T n = I n (; (2 J n e i = e i, e i, e ; (3 x C n, (I n J T n J n x = (x T e e Jordan 82

10 32 ( Jordan 28 A n P n n 2 n m, n + n n m = n, P AP = J n J n2 J nm (32 Jordan, A Jordan, J n j, A Jordan, (32,, n n 2 n m, n + n n m = n, (32 A n n =, A = (, < n, A ( α T A α C n A n, n P P A P (32,, A 2 = P A P = J k J k2 J ks, k k 2 k s, k + k k s = n P 2 = ( P, A 2 = J k J J = J k2 J ks ( A 3 = P2 α AP 2 = T α P T α T 2 = J A k, 2 J (α T α T 2 αt P α T J T k P 3 = I, I x C n, (I n J T n J n x = (x T e e ( 32(3 (α A 4 = P3 T e e T α T 2 A 3 P 3 = J k (322 J Jordan 87 83

11 α T α T e 2 J k =, A 4 = J k α2 T J J ( α T 2 Jordan J J A J k J, a = α T e, P 4 = (a I (ai, ( 7 A 5 = P 4 A 4 P 4 = ( J e α2 T J ( e J T = = J J k + 32(2, J e i+ = e i, k ( I e2 α T 2 I ( J e α T 2 J ( I e2 α2 T = I (323 ( J e 2 α2 T J (324 J, e α2 T e 2α2 T J, ( 8, (324 e 2 α2 T J N, N k J k =, A 5 J J, Jordan n n = < n, n A Jordan J = J n J n2 J ns J = J m J m2 J mt n n 2 n s, m m 2 m t n = m,, n > m, J m J m =,!, 253, J n2 J ns J m2 J mt,, Jordan, s = t n 2 = m 2,, n s = m s ( 6: 32 M N n M N r(m k = r(n k, k 32 λi n + J n n λ-jordan, J n (λ Schur Jordan Jordan 322 (Jordan A n, P n n 2 n m, n + n n m = n, P AP = J n (λ J n2 (λ 2 J nm (λ m (325 Jordan, A Jordan n j, A Jordan 84

12 (325 λ i 2 322, A A Jordan, Jordan,,, 32, 323 A t = ( t t, Jordan D = ( t t =, Jordan A = J 2 t, A t Jordan Jordan, Jordan ( Jordan,, Jordan,,,,, A M n(q P M n(q P AP Jordan? Q R? «A B 2 Jordan A, B Jordan?? B A A = A = B 3 ( 32? Jordan Jordan, Jordan,?? Jordan 33 ( Jordan A, Jordan Jordan A e, Jordan J, J Jordan f A e = J e =, f e, J f = A f =, e f 33, Jordan : A = A 2 = A, 2 33, A Jordan J Jordan 2 r(a = 2, A 2, 2, J = J 2 J 2 85

13 , Jordan,,, 2 33 n A Jordan J (32, e ( e = max{n i i m}; (2 J Jordan m A ; (3 J k Jordan l k, A k η k, k m (33 l k = 2η k η k η k+, k m (33 η =, η, η 2,, η k, η k, η k+,, l k = (η k η k (η k+ η k 2 (2 J Jordan A Jordan,, 332, Jordan : A = A 3 = A, 3 r(a = 5 33, A Jordan J 3, Jordan 3 3 2, J = J 3 J 3 J A 33, P, P AP = J P = (α, α 2, α 3, α 4, A(α, α 2, α 3, α 4 = (α, α 2, α 3, α 4 (J 2 J 2 = (, α,, α 3 P Aα =, Aα 2 = α, Aα 3 =, Aα 4 = α 3 (332, : b b 2 b 3 b 4 b + b 2 b 2 b + b 3 b + b 2 + b 4 86

14 Ax = β, β = (b, b 2, b 3, b 4 T b = b 3, b + b 2 + b 4 = { x x 3 =, x 2 +x 3 +x 4 =, α = (,,, T, α 3 = (,,, T Ax = β Ax = α, { x x 3 =, x 2 +x 3 +x 4 =, α 2 = (,,, T Ax = α 3, { x x 3 =, x 2 +x 3 +x 4 =, α 4 = (,,, T, P = (332 α α 3, α 2 α 4, (! A 2 x = (333, λ n A, (A λi k x = (k A ( λ, ( 2-24 ( (A λi k x = C n, A ( λ, E λ Jordan C n = E λ 2-24, λ σ(a Jordan λ- Smith 29, 56-6 Schur, 33 Jordan 332 n A Jordan J (325 µ A, (A µi k η k, J µ k Jordan l k, ( η J µ Jordan ; (2 l = 2η η 2, l k = 2η k η k η k+, k 2, ( 33, ( 3 n A ( A n = 33, 332 A,, 29 Henry John Stephen Smith( , - 87

15 334 Jordan J, P, P AP = J: 3 A = λi A = (λ 4, A Jordan J A I 3, 3, J = J 3 ( ( P AP = J P = (α, α 2, α 3, α 4, AP = P J A(α, α 2, α 3, α 4 = (α, α 2, α 3, α 4 (J 3 ( ( = (α, α + α 2, α 2 + α 3, α 4 : Aα = α, Aα 2 = α + α 2, Aα 3 = α 2 + α 3, Aα 4 = α 4, (I Aα =, (I Aα 2 = α, (I Aα 3 = α 2, (I Aα 4 = α = (,,, T, α 2 = (, 2,, T, α 3 = (, 3,, T, α 4 = (, 2,, T, P (? P = Jordan : A = A A λi A = λ 3 5 λ λ 3 = (λ 3 (λ + 3 (λ λ + λ λ + 2 λ 2 λ 2

16 3 A (, A 2, 2 A 3 Schur N i A B = (I 3 + N ( I 3 + N 2 (2I 2 + N 3, N i A i λ i I A I, N J 3, Jordan, A I A 2I N 2 N 3 J 3 J 2 A Jordan J = J 3 ( J 3 ( J 2 ( n A Jordan r(a = A Jordan A n, tr A A Jordan : ( diag (tr A,,, ( tr A, (2 ( tr A =, A, 2 Jordan Jordan? 2 P Q Jordan,? :, ( 825, Abel 3 Galois 3 5 5,,,, a i A i A n, A = (a ij n, D i (A = {x C x a ii j i i= a ij } (i =, 2,, n A i j i a ij R i (A, A, A i D i (A = {x C x a ii R i (A} 3 Niels Henrik (82-829,,, Abel Prize ( 3 Évariste Galois(8-832,, 89

17 , A n i= D i(a A ( Gerschgorin 32, G(A, G(A Gerschgorin, Gerschgorin 34 A, A = 2 i 2i D (A = {x x 2}, D 2 (A = {x x 2 }, D 3 (A = {x x 2i 2} 342, A = (a ij, A A = ( a ij 34 : ( 33 A n, λ n λ a ii a ij, i =, 2,, n j i, σ(a G(A, A A λ A, α = (x,, x n T Aα = λα, a x + a 2 x a n x n = λx, a 2 x + a 22 x a 2n x n = λx 2, ( a n x + a n2 x a nn x n = λx n max { x,, x n } = x m, x x m (* m (λ a mm x m = j m a mj x j λ a mm x m j m a mj x j x m j m a mj, λ a mm j m a mj λ m A T, A 32 Semyon Aranovich Gershgorin(9-933, ( 33 Gerschgorin 93 9

18 343 A, A = A λ = 6, λ = 8, λ = 8, λ = 6 : 4 3, n A ( a ii > j i a ij, i =, 2,, n, : A = A λ A,,, k, λ a kk j k a kj λ =, a kk j k a kj,! A, A,,,,, 345 A, A = ( 8 5 9

19 , A λ = 5 + 5i, λ 2 = 5 5i A λ 8, λ 5 : λ λ 2 342, λ 8, λ 5! : 34 ( C k, C k,, A = (a ij D, B = A D A(ε = D + εb, ε A( = D, A( = A A(ε ε, A(ε ε, ε, A(ε a ii, εr i (A ε, C k, C, A n k ε [, ], A(ε ε =, a, a 22,, a nn, k k, n k,, ε [, ],, ε =, 343, A 4, 3 ( 343 : ( λ + 4 6; (2 λ + 8; λ 6; (3 λ 3 8,, A,, (, 34 n A, A A, A 92

20 , ( 4 A = 2 x 4 x = 2, A x 4,, A, σ(a = σ(p AP (, A,,, P = D = diag (d, d 2,, d n, d i > B = D AD = (a ij d j d i B A, ( (d j : j, d j >, d k, B ;, j, d j <, d k, B, A, ( ( ( d B = 4 d d = 2 2 d 2 ( 4d 2 d, 2 B x 4d 2 d x = 2, 4d 2 d 2, 346 A, A = A : λ 9 2; λ 8 3; λ 4 3,,, D AD = B, D diag (,,, 9 B = B λ 9 2; λ 8 22; λ 4 3 B, A B,, A, 34, 347 A, A =

21 A D = λ 7 4; D 2 = λ 8 2; D 3 = λ 5 ; D 4 = λ,, G = D D2 D3 G 2 = D 4 G 2,,, G A 342 A = (a ij n ν = max n j= a kj ( k n, ρ(a ν λ A,, k λ a kk n j=,j k a kj λ a kk + n j=,j k a kj = n j= a kj ν, ν, ρ(a ν max j n A A T, A ν, ν = n k= a kj, A T 343 ρ(a min{ν, ν }, 93,,, 5 5, A = (a ij : C i (A = j i a ji, i n 342 ( Ostrowski 34 λ n A, α i n λ a ii R i (A α C i (A α i Ostrowski Ostrowski,, 348 A = ( 4 A x 4 x 6 6, α = 2, Ostrowski x 2 x 6 2, A,, x 2, Alexander Markowich ( ,, 94

22 343 ( Brauer 35 λ n A, i j n λ a ii λ a jj R i (AR j (A ( Cassini 36, 343,,, 52? :, Jordan Jordan ( Jordan Schur, Jordan,,, PCA : n ( {x,, x n } N { x,, x N },, (, {x,, x n } n ( C n { x,, x N } N ( C N,, ( n, ( ( (, n N (, N ( n 35 x = (x,, x n T (, x E{x} = (E{x },, E{x n } T ; x x 2 = x x E{ x 2 }, E x x R x = E{xx } = (E{x i x j } n n x R x Hermite, ( 56, λ λ 2 λ n N λ,, λ N x : : 35 Alfred Brauer( ,, Richard Brauer (University of North Carolina, Chapel Hill 36 Giovanni Domenico Cassini(625-72,,, Cassini 95

23 n {x,, x n } N x j = n ā ij x i = αjx, i= j =, 2,, N x = (x,, x n T, α j = (a j, a 2j,, a nj T : N, ( δ ij Kronecker ( x i, x j = x j x i = x α j α i x = x xα i α j = x 2 α i α j = δ ij, ( x, x 2 α i α j = δ ij x 2 : α i (i =, 2,, N x R x N v i, E xi = E{ x i 2 } = E{(αi x αi x} = E{α i x(α i x } = vi E{xx v i } = vi R xv i v = vi (v v 2, v 2,, v N diag (λ, λ 2,, λ N v i = λ i v N λ i (i =, 2,, N, E x E x2 E xn x i (i =, 2,, N R x x i E xi, R x N λ i (i =, 2,, N, tr(r x = E x + E x2 + + E xn λ + λ λ N E x + E x2 + + E xn = λ + λ λ N E x + E x2 + + E xn, n N, ( A, A, A > A n ( A, ( B C P P T AP =, B, C, D, A, D 96

24 ( A ( ( 62:, ( 35 A n, A (I + A n > ( 352 (Perron 37 -Froubenius 38 A n, ( A ρ(a A ; (2 A ρ(a ; (3 A λ Perron-Froubenius, (, [7] [4] Perron- Froubenius ρ(a A Perron, Perron A Perron Perron-Froubenius, Perron Perron Perron n E(,, A = (a ij n n, a ij i j a ij, n a ij =, j =, 2,, n (35 i= A ( A, A, A :?, P i i, P = (P,, P n T ( P P, P 353 (Leontief 39 A E, P n P P A Perron n j= a ij i j, i a ij P j P P i = n a ij P j, i =, 2,, n (352 j= 37 Oskar Perron(88-975,, Perron,,!( 38 Perron 97, Frobenius Leontief Input-Output Economics, Scientific Amrican,October 95, pp

25 AP = P,,, i γ i, γ i = P i n a ij P j, i =, 2,, n (353 j= γ = (γ,, γ n T γ, γ (353 (I AP = γ (354 γ ρ(a < ( 63 A,, (, r > γ i = rp i, (354 (I AP = rp (355 AP = λp, λ = r, P A, Perron-Froubenius P A Perron!,, Perron, Perron? 2? 3 P, A P, P T AP A? Schur 2 A 22 ( Sylvester A A 6 ; (2 A, 3 α = (a,, a n T, β = (b,, b n T, x, A = xi n + αβ T ( A ; (2 Sylvester A ; (3 A 4 α β, α a b c B α d e β xa β 98

26 5 ( Schur ; (2 Schur : A M n (R, Q Q T AQ = A A 2 A k «ai b i A i 2 (b b i a i i (:, λ A, Ax = λx, A x = λ x, x x, Rex Imx,, 6 a, V = {e ax f(x f(x C n [x]} n ( : α d α V ; d x (2 Jordan σ R 3, x = (x, x 2, x 3 T, σ(x = ( 2x 2 2x 3, 2x + 3x 2 x 3, 2x x 2 + 3x 3 T R 3, σ «9 B =, V = {X = (x ij M 2(R tr X = } σ σ(x = B T X X T B, X V V, σ A,, (, 2 T, (2, T A A : « ( ; 5 4 A; , ( ; (2 ; (3 C A A; 3 n A : ( A n k A k ; (2 A, A 4 n A,, A 5 AB = BA, A B? 6 ( 32; ( Jordan (323 8 Jordan a = α T e A Jordan 2 A = B A A 99

27 9 Jordan Fitting 4 ( (255: V n, σ EndV, V = Im(σ n Ker(σ n 2 33 (2 ( Jordan, λi A = n A, σ(a = {λ, λ 2,, λ s }, g i λ i 2 E λi = {x C n (A λi n i x = } 22 dim C E λi = n i, C n = P λ σ(a E λ ( s Q i= (λ λ i n i 23 α j E λi, j g i, j gi {α j, (A λ i Iα j,, (A λ i I m j α j } E λi ( α j, E λi A 24 C n Jordan A C n Jordan Jordan ( A x Ax , A P Jordan : ( 4 4 A ; ( A ; ( A ; ( A ; ( A A Jordan, P P AP = J: A ; 4 A ; (3 B A , : A 7 6 3A, B 4 4 2A, C 3 2A, D A ( ; (2 A A A 3 A A = D + N, D, N, DN = ND( Jordan-Chevalley 4 32 A A ( A A ; (2 A Jordan-Chevalley? 4 Hans Fitting(96-938, 4 Claude Chevalley(99-984,,,,

28 33 p(λ = ( n [λ n a n λ n a n 2 λ n 2 a λ a ] a n a n 2 a a C = C A p(λ n A ( n p(λ ( C ; (2 n = 2, A C A ; (3 (2 (: Frobenius 34 V e x, xe x, x 2 e x, e 2x V : T (f = f T Jordan Jordan 35 A, A Jordan? 36 ( Fourier σ C n, σ((x, x 2,, x n T = (x 2, x 3,, x n, x T : (σ λ n = λ j = e 2πi n j, j n; (2σ λ j α j = (λ j, λ 2 j,, λ n j T, α j = n; (3α,, α n C n ; (4 x = (x, x 2,, x n T P σ α j x = n P a jα j, x k = n (5 a j = (x, α j/n = n np k= x k e 2πi n jk ; (6 σ 7 Fourier, 9 37 A i A 3 38 A j= A A j= a je 2πi n j ; 39 A A ( A ; (2 A ( 4 Hilbert 42 A = n n n (n+ 2 2 n 2 3 n 3 4 n n (n+ 3 2n C A, A 42 David Hilbert( ,,, 9 23 Hilbert : We must know, we shall know

29 4 Ostrowski : A = C A 42 σ(a G(A G(A T A A 44 σ(a = P G(P AP, P P? 45 A = (a ij s, r(a s 46 n A = (a ij n,, A 47 Hadamard 43 Q : n A = (a ij A n P ( n a ij 2 /2 : ( A, A n Q a ii; i= (2 C, a ij C, i, j n, A C n n n/2 ; i= i= (3 a ij = ±, i, j n, (2 A n n/2, A Hadamard A Hadamard A T A = ni n A 48 A = (a ij, D = diag (a,, a nn D ρ(i D A < 49 Ostrowski Brauer 5 Ostrowski 5 Brauer 52, Brauer A = C A 53 a =, a =, a 2,, a n, a n+ = xa n + a n, n, a n, x ( x =, Fibonacci A n, y T A = λy T y A λ : A λ µ (λ µ 55 (, C C = d dx : : f(x f (x σ( 43 Jacques Salomon( ,,, Hadamard : k, 4k Hadamard 44 Leonardo Pisano Leonardo Bonacci(7-25,, 2

30 56-6 Jordan λ- ( λ,, n λ- ( r λ- A(λ ( ( A(λ Smith diag (d (λ, d 2(λ,, d r(λ,,,, d i (λ d i (λ d i+ (λ, i r d i (λ A(λ 57 A(λ, A(λ Jordan, 58 λi A λi B 59 A B λi A λi B 6 A Jordan 6? A, A, A A, (I AP = γ γ P ρ(a < (: (I A, ρ(a < 45 Henry John Stephen Smith ( , 3

4 A C n n, AA = A A, A,,, Hermite, Hermite,, A, A A, A, A 4 (, 4,, A A, ( A C n n, A A n, 4 A = (a ij n n, λ, λ,, λ n A n n ( (Schur λ i n

4 A C n n, AA = A A, A,,, Hermite, Hermite,, A, A A, A, A 4 (, 4,, A A, ( A C n n, A A n, 4 A = (a ij n n, λ, λ,, λ n A n n ( (Schur λ i n ,?,,, A, A ( Gauss m n A B P Q ( Ir B = P AQ r(a = r, A Ax = b P Ax = P b, x = Qy, ( Ir y = P b (4 (4, A A = ( P Ir Q,,, Schur, Cholesky LU, ( QR,, Schur,, (,,, 4 A AA = A A Schur, U U AU = T AA = A A