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 T T = T T A, A, U U AU = D, AA = (UDU (UD U = UDD U = (UD U (UDU = A A A = (a ij, a ij =, i > j AA = A A,, a a + a a + + a n a n = a a, A A, : 4

4 A C n n, AA = A A, A,,, Hermite, Hermite,, A, A 4 4 4 A, A, A 4 (, 4,, A A, ( 4 4 4 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 ij ; i= i= i,j= ( A n λ i = n a ij i,j=, 9 Schur U AU = B, (, (5 n tr(aa = a ij i,j= tr(aa tr(bb 4 A, A = A, 4 A =? 44 (,, ( 5

, 4 Schur a b b ( a b b a (4 8, Gauss, (4 Schur a + bi, ACayley( 845, Schur a ± bi 45 (4 Schur A b, A a ± bi, (a + bi (a bi A A = AA = (a + b (a + b Schur, ( 4 5, 44 ( A n, A Q A i, Schur Q T AQ = A A A s (4 Hermite (,, (,? (4,, 4 A n ( A Q Q T AQ k tr(a k = n i= λk i, λ,, λ n A ; ( A Q Q T AQ = A A A s ( bi A i =, ; b i ( A Q Q T AQ = I s ( I t A A A m A i Givens, B n (4 B Hermite U U BU ; (5 A Hermite U U BU ( ; (6 A U U BU, ; (7 Hermite A A 6

4, (7 4 Hermite A k A k (, A k V k λ, λ,, λ m A, λ m+, λ m+,, λ n α, α,, α n λ, λ,, λ n W = Span{α m+, α m+,, α n } α m+, α m+,, α n x W, x Ax (? m < k, ( dim(v k + W dim C n = n, dim (V k W = dim V k + dim W dim(v k + W k + (n m n = k m > V k W {} x V k W, x Ax >! m k, A k 46 ( Hermite A n Hermite, x Ax, A ( λ y + λ y + + λ n y n (4 λ i, Hermite A,, ( R,,, R x 4 + y 4 + z 9 =, xoy- 47 ( 44, A ( a b Schur, b ( b a,, Schur a + b,, σ σ ( (??? 4 45? 5 46, ( 7

A, 4, U U AU = diag (λ, λ,, λ n A = U diag(λ, λ,, λ n U U = (α, α,, α n, λ λ A = (α, α,, α n λn = λ α α + λ α α + + λ nα n α n (4 λ, λ,, λ n A, α, α,, α n, (4 A ( (4 (,, (4 λ, λ,, λ s A α α A = λ P + λ P + + λ s P s, (4 (α i α i = α i α i, i n, (α i αi (α jαj =, i j n, (α i αi = α i αi, i n, P i = P i, P i = P i, P i P j =, i j s (4, P i (, 4 ( A λ, A = λi A λ λ, A = λ P + λ P, α R, Aα = λ P α + λ P α (44 (P α, P α = (P α T P α = α T P T P α =, λ P α λ P α (44 Aα, n (4 α n 4 A = ( A ( = i ( i i i ( i i, A ( x x ( x ix = i ix + x ( x + ix i, ix + x, 8

4 A = 8 4 4 7 4 4 8, A λi A = λ 8 4 4 λ + 7 4 4 λ 8 = (λ 9 (λ + 9 A λ = 9(, λ = 9 : α =, α =, α = 4 P = α α + α α = (,, + (,, = 4 4 + = 7 4 4 4 9 8 4 4 4 7 P = α α = 4 ( 6,, 6 = 4 4 6 4 8 4 A A = 9P 9P 44 A = (, A T A AA T A T A =, λi A T A = λ(λ (λ, : ( (,, T (,, T A T A A T A = [ 9 ] + [ 6 4 ]

, AA T AA T = [ ( ] + [ ( ] (45 45 A Hermite, A A n A = λ i α i αi, i= ( A = n i= λ i α i α i (46 44 AA T, (45 (46 (AA T = [ ( ] + ( [ ] = ( n A(, A A s λ, λ,, λ s, k, k,, k s α i, α i,, α i,ki λ i P = (α, α,, α,k, α, α,, α,k,, α s, α s,, α s,ks, k {}}{{}}{{}}{ P AP = diag( λ,, λ, λ,, λ,,, λ s,, λ s, k k {}}{{}}{{}}{ A = P diag( λ,, λ, λ,, λ,,, λ s,, λ s P k k s k s P = β T β T,k β T s β T s,k s A = s λ i (α i βi T + α i βi T + + α i,ki βi,k T i (47 i=

P i = α i β T i + α i β T i + + α i,ki β T i,k i, i =,,, s (48 A = λ P + λ P + + λ s P s (49 (49 A P P = I, s (α i βi T + α i βi T + + α i, ki βi, T k i = I, i= s i= P i = I P P = I, βij T α kl = δ ik δ jl P i P j =, i j, (α ij βij T = α ij βij T, P i = P i k {}}{{}}{{}}{ A T = (P T diag( λ,, λ, λ,, λ,,, λ s,, λ s P T, k k s, k {}}{{}}{{}}{ A T (P T = (P T diag( λ,, λ, λ,, λ,,, λ s,, λ s, k k s (P T A T λ,, λ, λ,, λ,, λ s,, λ s (P T = (β,, β, k, β,, β, k,, β s,, β s, ks, A T β ij = λ i β ij, β T ija = λ i β T ij, i =,,, s; j =,,, k i (4, β T ij A (, α ij A s i= P i = I, n = r(i s r(p i k + k + + k s = n, i= k i λ i, i =,,, s, (48 r(p i = k i, ( 9: 4 ( A n, A σ(a = {λ, λ,, λ s }, λ i k i s n P, P,, P s : ( A = (4 s i= λ i P i ; ( P i = P i ; ( P i P j = ( i j ; s P i = I; (5 r(p i = k i i=

P i A (,, Hermite, Ax = λ P x + λ P x + + λ s P s x P i x 46 : A = 5 λi A = λ + λ 5 λ = (λ (λ A λ =, λ = ( (λ i I Ax =, λ, λ α = ; α =, α = P = (α, α, α, P = 4 = β T β T β T, P = α β T = P = α β T + α β T = = 4 4 (,, = + (, 4, +, = (,, 4 A A = P + P, P P, P e P e 4 A = s i= λ ip i A, A m = s λ m i P i (4 i= f(x f(a = s i= f(λ ip i

47 A 46, e A = n= n! An 4 (,( (4, A n = (P + P n = P + n P, n e A = n= n! An = ep + e P 4, A λ, A? n A B AB = BA =? 4, Cholesky Ax = b A (, (, A, A = LU, L U, y = Ux Ly = b Ux = y AX = B,, L( U (, 4 A n, U L A = LU (4 A LU 46, (4 A LU 4 ( A = (,, 4 ( LU A n, A A = LU, A = U = u u u nn A = LU = L U A, A,, L L = U(U, L L, L L L = L, U = U,, 46 LU Alan Mathison Turing ( 948

4 ( Cholesky 47 A A = LU, G A = GG T ( Cholesky, G Cholesky A n n = A, a >, a a G = a a a n a ( G AG T a =, A A n (? G A, a i > G AG T = diag (a, a,, a n A = G diag (a,, a n G T = G diag ( a,, a n diag ( a,, a n G T, A = GG T, G = G diag ( a,, a n 4 A = ( Cholesky 4, A, ( G =, G = G G AG T = ( ( = ( G, A = GG T a = g, a = g g, a = g + g, g =, g =, g = 47 André-Louis Cholesky(875-98,, 4

Cholesky, A n, A = GG T j a ij = g ik g jk, i j k= j g jj g ij = a ij g ik g jk (4 v(i, ( v(i, (4 i = j gjj = v(j, g ij = v(i = v(i (4 g jj v(j (4 (4 Cholesky, Cholesky G A, G = A k= 4 A Cholesky A = GG T, A A = G T G (44 4, 5 4 n A A A, L U D, A = L DU, D A, 4 n A r(a, A LU, UL(,? LU UL,? Cholesky?? 4, Cholesky? 5? QR,,,, Gram- Schmidt 44 A C n n, A, U R A = UR (44 5

α j A j, j =,,, n A = (α, α,, α n A, α, α,, α n Gram-Schmidt α, α,, α n η, η,, η n : η = α, η = α (α, η (η, η η, η = α (α, η (η, η η (α, η (η, η η, η n = α n (α n, η (η, η η (α n, η (η, η η (α n, η n (η n, η n η n α = η, α = (α, η (η, η η + η, α = (α, η (η, η η + (α, η (η, η η + η, α n = (α n, η (η, η η + (α n, η (η, η η + + (α n, η n (η n, η n η n + η n β i = η (ηi i, U = (β, β,, β n,,η i α = (η, η β, α = (α, β β + (η, η β, α = (α, β β + (α, β β + (η, η β, α n = (α n, β β + (α n, β β + + (α n, β n β n + (η n, η n β n A = UR, U = (β, β,, β n, R = (η, η (α, β (α n, β (α n, β (η, η (α n, β (α n, β (ηn, η n (α n, β n (ηn, η n A = U R, U, R UR = U R RR = U U,, RR, ( U = U, R = R, 6

(44 A, UR A, U, Q, R, UR QR 44 A A = UR, R A, 44 A C n r, A, A = UR, U C n r r, R C r r, 44, 6 A = UR = U R, U, U, R, R U U = I r, U U = R R, (!U U, R R! 44, U n, R n r ( R, R =, r R A, 44 A QR 44 U U = I r,, U U U = I, U, 44 44 ( 7,, ( 44 A C m n A = UR, A A = R R, R A A Cholesky A = RU, R, U,??? 4 U U = I UU = I?,, (,,? A A = UDU,,?,, Beltrami 48 87 48 Eugenio Beltrami(85-9,,,,, 7

f(x, y = x Ay x = Uξ, y = V η, U, V, f(x, y = ξ U AV η, D = U AV, A = UDV, D m n, diag (d, d,, d n m n m n ( m n d d d d n, m n, diag (d, d,, d m m n m n 45 ( A = (a ij C m n (m n, σ σ σ r >, m n U V, A = UDV (45 D = diag (σ, σ,, σ r,,, m n (45 A, SVD, σ,, σ r,,, ( n A, A, A A, σ(a A = {σ, σ,, σ n}, σ σ σ r > = σ r+ = = σ n A A, α, α,, α n A A C n V V = I r, V = (α, α,, α r, V = (α r+,, α n, Λ = diag (σ, σ,, σ r A AV = V Λ Λ V A AV Λ = I (45 A AV = AV = (? U = AV Λ, (45 U U = I, U, U = (U, U V = (V, V, ( U U AV = AV U AV ( U AV U AV = Λ U U Λ = ( Λ = D, A = UDV, m < n (, A σ max (A σ min (A σ σ n, Av i = { σi u i, i r, r + i n (45 8

u i A = { σi vi, i r, r + i m (454 V U A, V U A A, (45 (454 A A v i, σ i u i = Av i, i n( (45 (454 u i AA, (45 (454, u i, i > n Hermite ( 45 : A = (, A A AA AA = ( = ( AA λi AA = λ λ = (λ (λ AA λ =, λ = β = (, β = ( A A λ =, λ =, λ =, α = A β = λ, α = A β = λ 6, α = α Hermite V = (α, α, α, U = (β, β, D = (,, A = UDV = ( ( 6 6 6 9

45 A, A A A = U ΛU, Λ = diag (λ, λ,, λ n W D = ΛW = diag ( λ, λ,, λ n, A = U ΛW W U = U DW U = U DV, V = W U (?, A A = U DV 45 ( A m n = UDV C n C m, ( C n x V x, ( C n V x r = r(a ( (C m DV x, ( C m UDV x, 454 ( S : x + y = A = ( 5 A T A =, 8, A 5 A = ( ( ( (, A S :,, S ;,, S,, ( A (45 A = n σ i u i vi (455 i= A σ j =, j > r, (455 r A = σ i u i vi = U r D r Vr (456 i= U r, V r U, V r, D r = diag (σ, σ,, σ r A, ( 6 Sylvester, 45 A = (a ij m n r, AA A A ( σ, σ,, σ r, σ, σ,, σ r A, A A

455 A A = UDV, A = V D U, A σ σ σ n >, A /σ n /σ n /σ > A (, Cond(A, Cond(A = σ (A/σ n (A (457 A = U DV A, P = U DU, U = U V, 45 A C n n, U P A = P U (458 A P U A Hermite ( 7 A P = AA, P = AA ( AA, (AA / A, U 456 ( z = re iθ, r = P, e iθ = U, A = re iθ A, θ, r, ( 8 45 A = P U A, A P U = UP,,, 4-4 45 (, 45 ( A = UDV m n A, r = r(a, ( U r A ; ( V r A ; ( U m r A ; (4 V n r A 454 ( λ n A, σ max (A λ σ min (A,

( 457 A =, A, ( ± 5/ 458 ( A C m n, tr(a A = r i= σ i, 4 Schur, 455 ( A A, A A 459 A C m n A = A A =, A =, ( A /m?,?? U P A = UP??? 4 B = A B A?, B m = A B :,, QR QR A A = UR, Ax = b A Ax = A b, R Rx = R U b, Rx = U b, x = R U b, A = QR, x = R Q T b 46 QR AX = b, A = 5 8, b = 7 9 A α, α, α, : η = α = (,,, T, η = α (α, η (η, η η = (,, 5, T (,,, T = (,,, T, η = α (α, η (η, η η (α, η (η, η η = (,, 8, 7 T (,,, T 6 6 (,,, T = (,,, T

, β =, β = 6, β = Q = (β, β, β Q T Q = I, A = QR R = Q T QR = Q T A = = 6 6 6 6 6 6 5 8 7 R = 6 6 x = R Q T b = = 6 6 6 6 6 6 6 6 6 = 9 x = (, T, 46 ( ( x = A = (, 44, A = R Q T = (, x = = QR, (

QR QR A = A = Q R n A QR A m+ = R m Q m A (, :!, A, A,, A k, (46 R (46 A, R A, 46 5 5(, 6 n n A A = UDV T A, D ( A m, σ,, σ m u i v i ( i m, m + mn + mn = m(n +, n n /(mn + m (, Â = m σ i u i vi T (46 i=, m, m, m, 5, m = 5, 5( 5 + = 55, 55, ( 46 A, B, C = C = UDV, U ( ( ( A = UDV U U = Dr V, B U U ( A U = DV B, ( A B U A = U B (46, A R(A B R(B, ( 48 R(A R(B = R((U A = R(A U, R(A R(B 4

,, U, U AU : i i i i ( A = @ i i A; ( A = @ i A; ( A = @ i i i i? 4 4 5 A n, x (A xi ; ( x, Ax A x ; (A A ; (4A 6 A, ( A Hermite A ; ( A A ; ( A A ; (4 A λ, λ,, λ n, AA A A λ, λ,, λ n? 7 A, ( A, A Hermite ; ( A = A, A = A; ( A Hermite, ( A k = I, A ( A = I 8 : A Hermite, λ λ λ n, : λ k = min w i C n i n k max x w i i n k x Ax x x = max w i C n i n k x Ax min x w i x x i n k A, 9 4 λ max = λ n = λ min = λ = x Ax max x,x C n x x = max x x= x Ax, x Ax min x,x C n x x = min x x= x Ax (, 4 4 ( 47 ; «(? «(, ; (4 4 P, Q m n, : m + n A = B 5 P B Q «, P, Q,

5 Sylvester, Hermite, ( 9 @ 6 A = @ A, 7 S, C, C SC = @ 7 4 4 4 8 4 8 A, C 8 9 ( 4 45 A S = A = LU, L U, L U A = L U «A = Cholesky ( 4 4 Hermite Cholesky 5 4 6 44 7 A = @ A, b = @ 6 8 ( R(A ; ( A QR ; ( Ax = b ; (4 u = (,, T, u = (,, 8 QR : ( A = @ A T, R(A, R(A A A; ( A = @ A; ( A = @ 9 A, B C m n : A A = B B U B = UA 8 A C m n, : σ min (A = min{(x A Ax /, x x = }, σ max (A = max{(x A Ax / : x x = } A σ : σx = x a(x, ww, x R n, w, a, σ? w,? A C m n r >, A A = U diag(s,, s r,,, V, B = 6 A A «

4 454, A 5 ( ; ( Jordan J n(λ 6 45 7 n A A = P e ih, P, H Hermite 8 45 9, x C n 4 45 4 45, A = @ 4 454 4 458 A 44 x, y, α, β C n, x y = α β = A = xα + yβ, A F- 45 A Hermite P P AP 46 (46 A = A, A, A A,? 47,, 48 46 (, 9 49 A, B C m n : x N(A N(B A B «x = 5 : A C m n σ σ σ n, :, σ k = min w i C n i n k σ max = σ = σ min = σ n = Ax max x w i i n k x = max w i C n i n k Ax min x w i x i n k Ax max = max x,x C n x Ax, x x= Ax min = min x,x C n x Ax x x= 5 : A, B σ max(a + B σ max(a + σ max(b 5 ( A m n r, A = UDV, U = (u,, u m, V = (v,, v n k < r, A k = kx σ iu ivi, k < r i= : min A B = A A k = σ k+, k < r r(b=k min A B F = A A k F = σk+ + σk+ + + σr r(b=k 5, 7

54 ( A, B m n U, V D, Λ A = UDV, B = UΛV A B AB? 8