1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2 λ a nn λ n λ n 1
(λe n A)x = (λe n A) det(λe n A) = λe n A = λe n A = λ ( ) A λe n A = rank(λe n A) < n (λe n A)x = x ( II 4 ) x λ ( ) n n n ( ) n I 2 2 3 A = 1 1 1 1 3 1 ( ) Th9-1 λe n A = λ 2 2 3 λe 3 A = 1 λ 1 1 1 3 λ + 1 1, -2, 3 λ 3 2λ 2 5λ + 6 = λ 3 2λ 2 5λ + 6 = (λ 1)(λ + 2)(λ 3) λ = 1 (λe n A)x = λ = 1 1 2 3 1 1 x = 1 3 2 1 1 1 1 2
x = (x 1, x 2, x 3 ) T 1 1 x 1 1 1 x 2 = x 3 x = (1, 1, 1) T Ax = x ( ) x = (1/ 3, 1/ 3, 1/ 3) T λ = 2, 3 II( ) n ( ) 5 3 A = 3 1 λ 2 4λ + 4 = (λ 2) 2 = 2 1 ( ) (λe 2 A)x = ( ) 3 3 x = 3 3 x = ( ) 1/ 2 1/ 2 2 Th9-2 ( ) Ax = λx x T A T = λx T A A T = A x T A = λx T ( ) 3
n Th9-3 ( ) Th9-3 ( ) x y (x, y) (x, y) = x T y = x 1 y 1 + x 2 y 2 + + x n y n (x, y) = x T y = x 1y 1 + x 2y 2 + + x ny n (2) x i y i x, y z = a + bi z = a bi x x A A y, z x ( ) 1 (z ) = z 2 (y ± z) = y ± z 3 (yz) = y z 4 (y/z) = y /z 5 z z = z 6 z z = z 7 z z = zz 8 (zx) = z x 9 (x, x) x = 4
: (x, x) = x 1x 1 + x 2x 2 + + x nx n 7 Th9-3 λ x ( λ x ) x y (x, y) = x T y (Ax, y) = (λx, y) = λ (x, y) y = x (Ax, x) = λ (x, x) (3) (x, Ax) = (x, λx) = λ(x, x) (4) A A A = A A T = A (Ax, x) = (Ax) T x = x T A T x = x T Ax = (x, Ax) (3)(4) λ (x, x) = λ(x, x) x x (x, x) λ = λ 5 λ 5
A = ( 3 6 ) 6 2 λ 3 6 λe 2 A = 6 λ + 2 λ 2 λ 42 = 2 7,-6 λ = 7 (λe 2 A)x = (7E 2 A)x = ( ) 4 6 x = 6 9 ( ) 3 2 ( ) 1 3/2 x = λ = 6 ( ) 9 6 (λe 2 A)x = ( 6E 2 A)x = x = 6 4 ( ) 2 λ = 7 3 3 6
n A A T A T = A ( a ji = a ij) A A A n A A T A A T A = E n A A Th9-4 1 λ = 1 A Ax (Ax, Ax) = x T A T Ax = x T E n x = x T x = (x, x) A λ (Ax, Ax) = (λx, λx) = λ T x T λx = λ T λx T x = λ λ(x, x) (x, x) λ λ = λ 2 = 1 Th9-5 7
Th9-6 A n A λ 1, λ2 (λ 1 λ 2 ) x 1, x 2 A A T = A λ 1 = λ 1, λ 2 = λ 2 (Ax 1, x 2 ) = λ 1(x 1, x 2 ) = λ 1 (x 1, x 2 ) (Ax 1, x 2 ) = (x 1, A T x 2 ) = (x 1, Ax 2 ) = λ 2 (x 1, x 2 ) λ 1 (x 1, x 2 ) = λ 2 (x 1, x 2 ) (λ 1 λ 2 )(x 1, x 2 ) = λ 1 λ 2 (x 1, x 2 ) = 4 Th9-7 A k k 2 λ 1, λ 2,, λ k x 1, x 2,, x k x 1, x 2,, x k x 1, x 2,, x i 1 x 1, x 2,, x i i (2 i k) x i x 1, x 2, x i 1 x i = c 1 x 1 + c 2 x 2 + + c i 1 x i 1 (5) 8
(5) A Ax i = c 1 Ax 1 + c 2 Ax 2 + + c i 1 Ax i 1 x i A λ i (5) λ i (6), (7) λ i x i = c 1 λ 1 x 1 + c 2 λ 2 x 2 + + c i 1 λ i 1 x i 1 (6) λ i x i = c 1 λ i x 1 + c 2 λ i x 2 + + c i 1 λ i x i 1 (7) = c 1 (λ 1 λ i )x 1 + c 2 (λ 2 λ i )x 2 + + c i 1 (λ 2 λ i )x i 1 (8) x 1, x 2,, x i 1 (8) c j (λ j λ i ) = (j = 1,, i 1) (λ j λ i ) c j = (j = 1,, i 1) (5) x i = x i ( ) n A S S 1 AS Th9-8 A = S 1 AS (9) S 1 Ax = λx (9) S 1 Ax = λs 1 x x = Sy S 1 ASy = λs 1 Sy A y = λy (1) x A y = S 1 x A λ 9
Th6-8 Th9-9 n A P P 1 AP (A P ) n ( ) P 1 AP λ 1 λ 2 P 1 AP = Λ = λ n A = P ΛP 1 P 1 P 1 A = ΛP 1 (11) Λ Λ λ i e i Λe i = λ i e i x i = P e i e i = P 1 x i Λe i = λ i e i (11) ΛP 1 = P 1 A P ΛP 1 x i = λ i P 1 x i P 1 Ax i = λ i P 1 x i = P 1 (λ i x i ) Ax i = λ i x i x i = P e i A Λ λ i A e 1, e n P e 1,, P e n n ( {a 1,, a n } P {P a 1,, P a n } c 1 P a 1 + + c n P a n = P (c 1 a 1 + + c n a n ) = P 1 c 1 a 1 + + c n a n = {a 1,, a n } c 1 = = c n = ) 1
Th9-1 n A n x 1, x 2,, x n ( x 1 x 2 x n ) P P 1 AP x j λ j P 1 AP = P 1 A ( x 1 x 2 x n ) = ( P 1 Ax 1 P 1 Ax 2 P 1 Ax n ) = ( λ 1 P 1 x 1 λ 2 P 1 x 2 λ n P 1 x n ) P 1 x j P 1 P j P 1 x j = e j ( ) P 1 AP = ( λ 1 e 1 λ 2 e 2 λ n e n ) = λ 1 λ j λ n 5 51 n A P P 1 AP = c 1 Th9-7, Th9-1 A n P P 1 AP n n A A 11 c n
( n ) λe 3 A = 2 1 A = 2 4 1 2 2 1 λ 2 1 2 λ 4 1 2 2 λ 1 A 1,2 2 = (λ 1)(λ 2) 2 = λ = 2 x = (x 1, x 2, x 3 ) T 2 2 1 x 1 2 2 1 x 2 = 2 2 1 x 3 2 2 1 2 2 1 2 2 1 2 2 1 2x 1 2x 2 x 3 = x 3 = 2x 1 2x 2 x 1 1 x 2 = x 1 + x 2 1 x 3 2 2 2 1 x 1 =, x 2 = 2 1 2 12
λ = 1 x = (x 1, x 2, x 3 ) T 1 2 1 x 1 2 3 1 x 2 = 2 2 x 3 1 2 1 1 2 1 1 1 2 3 1 1 1 1 1 2 2 A 1 1 x 1 1 1 x 2 = x 3 x 1 1 x 2 = x 3 1 x 3 1 1 1 P = 1 1 1 2 2 2 2 1 2 1 1 1 1 P 1 AP = 1 2 1 2 4 1 1 1 = 2 2 3 1 2 2 1 1 2 2 2 x 1, x 2 a 1 x 1 + a 2 x 2 A(a 1 x 1 + a 2 x 2 ) = λ(a 1 x 1 + a 2 x 2 ) Gram-Shmidt 4 1 A = 1 2 2 1 2 13
λe 3 A = λ 4 1 1 λ 2 2 1 λ 2 = (λ 2)(λ 3) 2 = A 2,3 3 λ = 3 x = (x 1, x 2, x 3 ) T 1 1 x 1 1 1 x 2 = 2 1 1 x 3 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 x 1 1 1 x 2 = x 3 x 1 1 x 2 = x 3 1 x 3 1 x 1 1/ 3 x 2 = 1/ 3 x 3 1/ 3 ( ) 1 52 Schur n n Th9-11 n A Q Q 1 AQ = Q T AQ 14
Q λ 1 Q 1 AQ = Q T AQ = = R λ n R A Schur 1 A λ 1 u 1 Au 1 = λ 1 u 1, u 1 = 1 u 1 {q 1,, q n } q 1 = u 1 u 1 Gram-Schmidt q 1,, q n n Q (1) = ( q 1 q 2 q n ) Q (1) Q (1) Q (1) T = Q (1) 1 q 1 = u 1 Aq 1 = λ 1 q 1 AQ (1) = ( Aq 1 Aq 2 Aq n ) = ( λ 1 q 1 q 2 q n ) = Q (1) ( λ1 A (1) ) Q (1) T Q (1) T = Q (1) 1 Q (1) T AQ (1) = ( λ1 A (1) ) (12) 2 A (1) A (1) λ 2 u 2 u 2 n-1 15
u 2 q 1 = u 2 {q 1,, q n 1} q 1,, q n 1 n-1 Q = ( q 1 q n 1 ) Q Q T A (1) Q = ( λ2 A (1) ) n Q (2) Q (2) = 1 Q ( )Q (2) n (12) Q (2) Q (1) T AQ (1) Q (2) = λ 1 A (1) 1 Q = λ 1 A (1) Q Q (2) T Q (2) T Q (1) T AQ (1) Q (2) = = 1 λ 1 Q T Q T A (1) Q λ 1 A (1) Q 16
= λ 1 λ 2 A (2) (13) 1 Q (1) Q (2) Q = Q (1) Q (2) Q 1 = Q T (13) Q T AQ = Q 1 AQ Q 2 λ 2 A (1) A A (13) λ 1 λ 2 A (2) B λ λ 1 λ λ 2 λe n B = λe n 2 A (2) = (λ λ 1 )(λ λ 2 ) λe n 2 A (2) = (14) λ 2 B A A (1) λ 2 A A (1) u 2 A λ 2 3 1,2 Q = Q (1) Q (2) Q (n 1) Q 1 AQ = Q T AQ = λ 1 λ n 17
A n Q 1 AQ = det Q 1 AQ = det A = λ 1 λ n λ 1 λ n = Π n i=1λ i trace ( n A (i,j) a ij tracea = n i=1 a ii )trace traceab = traceba traceq 1 AQ = tracea = trace λ 1 λ n n = λ i i=1 53 Th9-3 A Th9-11 Q (1) A (1) Th9-11 A Q Q 1 AQ = Q T AQ = λ 1 λ n A A T = A (Q T AQ) T = Q T A T Q = Q T AQ 18
Q T AQ Q T AQ Q T AQ Th9-12 A Q Q 1 AQ = Q T AQ n n Th9-6 λe 3 A = 2 1 1 A = 1 2 1 1 1 2 λ 2 1 1 1 λ 2 1 1 1 λ 2 λ = 4 x = (x 1, x 2, x 3 ) T 2 1 1 1 2 1 x = 1 1 2 1 1 1 1 x = = (λ 1) 2 (λ 4) 19
x 1 = x 3, x 2 = x 3 1/ 3 1/ 3 1/ 3 λ = 1 x = (x 1, x 2, x 3 ) T 1 1 1 1 1 1 x = 1 1 1 1 1 1 x = x 3 = x 1 x 2 λ = 1 x 1 1 x 2 = x 1 x 3 1 + x 2 1 1 λ = 1 1 1, 1 1 Gram- Schmidt λ = 1 1/ 2 1/ 6 1/, 2/ 6 2 1/ 6 1/ 3 1/ 2 1/ 6 Q = 1/ 3 2/ 6 1/ 3 1/ 2 1/ 6 2
A n A k A k n A P P 1 AP = diag[λ 1,, λ n ] λ i (i = 1,, n) A P 1 AP P 1 AP = P 1 A 2 P = diag[λ 2 1,, λ 2 n] P 1 A k P = diag[λ k 1,, λ k n] A k A k = P diag[λ k 1,, λ k n]p 1 Fibonacci Fibonacci Fibonacci F =, F 1 = 1,, F n+1 = F n + F n 1 F n F n+1 /F n (1 + 5)/2 u i = u i = ( ) Fi+1 F i ( ) Fi+1 F i, u = ( ) Fi + F i 1 = = F i ( ) F1 F = ( ) 1 ( ) ( ) 1 1 Fi = Au i 1 1 F i 1 u n = A n u ( ) 1 1 A = 1 λ 1 det(λe 2 A) = 1 A 1 λ = λ2 λ 1 λ 1 = 1 + 5, λ 2 = 1 5 2 2 21
λ 1, λ 2 v 1, v 2 ( ) ( ) λ1 λ2 v 1 =, v 2 = 1 1 p8, Th9-6 u u T v 1 = λ 1, u T v 2 = λ 2 u v 1, v 2 u = αv 1 + βv 2 u T v 1 = α(1 + λ 2 1), u T v 2 = β(1 + λ 2 2) α = λ 1, β = λ 2 1 + λ 2 1 1 + λ 2 2 1 = λ 1 λ 2 α = λ 1 1 + λ 2 1 = λ 1 λ 2 1 λ 1 λ 2 = 1 = 1 λ 1 λ 2 5 β = 1 λ 1 λ 2 = 1 5 u = 1 5 (v 1 v 2 ) v i A ( ) Fn+1 u n = = A n u = 1 (λ n 1 5 F n ( ) λ1 λ n 2 1 ( ) λ2 ) 1 F n = 1 {( 1 + 5 ) n ( 1 5 ) n } 5 2 2 F n+1 F n = λn+1 1 λ n+1 2 λ 2 1 λ n 2 = λ 1 ( λ 2 λ 1 ) n λ 2 1 ( λ 1 + 5 2 λ 1 ) n 2 ( ) A n F n 22
6 Cayley-Hamilton f(x) = a m x m + + a 1 x + a n A f(a) = a m A m + + a 1 A + a E n A λ x Ax = λx, A 2 x = A(Ax) = A(λx) = λ 2 x, A m x = λ m x f(a) = (a m λ m + + a 1 λ + a )x = f(λ)x x n f(a) f(λ) f(x) A φ A (x) = det(xe n A) Th9-13(Cayley-Hamilton ) φ A (A) = O n ( ) A λ i φ A (λ i ) = (i = 1,, n) A Q Q 1 AQ = diag(λ 1,, λ n ) Q 1 AQQ 1 AQ = Q 1 A 2 Q = diag(λ 2 1,, λ 2 n) Q 1 A m Q = diag(λ m 1,, λ m n ) Q 1 φ A (A)Q = diag(φ A (λ 1 ),, φ A (λ n )) = O n φ A (A) = Q n Q 1 = O n Cayley-Hamilton xe n A G(x) = adj(xe n A) G(x) (j,i) xe n A i j n-1 (xe n A)G(x) = det(xe n A)E n = φ A (x)e n (15) p G(x) xe n A n-1 x at most n-1 G(x) x G(x) = C n 1 x n 1 + C n 2 x n 2 + + C 1 x + C (16) C i (i =,, n 1) n φ A (x) = det(xe n A) n n φ A (x) = x n + b n 1 x n 1 + + b 1 x + b (17) 23
(16) (15) (xe n A)G(x) = (xe n A)(C n 1 x n 1 + C n 2 x n 2 + + C 1 x + C ) = C n 1 x n + C n 2 x n 1 + + C 1 x 2 + C x AC n 1 x n 1 AC 2 x 2 AC 1 x AC = C n 1 x n + (C n 2 AC n 1 )x n 1 + + (C AC 1 )x AC (18) (15) (17) (x n + b n 1 x n 1 + + b 1 x + b )E n = E n x n + b n 1 E n x n 1 + + b 1 E n x + b E n (19) x k b E n = AC b 1 E n = C AC 1 b 2 E n = C 1 AC 2 b n 1 E n = C n 2 AC n 1 E n = C n 1 E n, A, A 2,, A n b E n = AC b 1 A = AC A 2 C 1 b 2 A 2 = A 2 C 1 A 3 C 2 b n 1 A n 1 = A n 1 C n 2 A n C n 1 A n = A n C n 1 = A n + b n 1 A n 1 + + b 1 A + b E n = φ A (A) O n φ A (A) = O n 24
7 1 (1) ( ) 2 1 1 2 (2) ( cos θ ) sin θ sin θ cos θ (3) 1 2 1 3 2 1 2 1 (4) 1 1 1 1 (5) 1 1 1 (6) 5 1 2 2 1 1 1 4 12 1 (7) ( e n e 1 e 2 e n 1 ) 2 3 A 4 A n n (a) A u 1,, u n A n n x A x 2 = (x, x) = 1 x x = a 1 u 1 + + a n u n 2 i a i = (u i, x) ii n i=1 a 2 i = 1 (b) A n µ i (i = 1,, n) µ max 1 n x 2 = (x, x) = 1 n x x T Ax µ max 5 A (3,3) A,1,2 (a) det(a T A) (b) E 3 + A E 3 E 3 + A 25
(c) B B = c 1 A + c 2 E 3 c 1 c 2 c 2 1 + c 2 2 = 1 B (c 1, c 2 ) 6 A n A λ i (i = 1,, n) u i (i = 1,, n) λ n λ n 1 λ 2 λ 1 x (u 1, x ) x k, y k x 1 = Ax, y 1 = x 1 / x 1 x 2 = Ax 1, y 2 = x 2 / x 2 x k = Ax k 1, y k = x k / x k y k k u 1 26