a n,1 a n,2 a n,j a n,p

Transcription

1 K R C p n K K n p K n p K 1, n 1, p K (a i,j ) 1in 1jp n p n p a 1,1 a 1,2 a 1,j a 1,p a 2,1 a 2,2 a 2,j a 2,p.... a i,1 a i,2 a i,j a i,p.... a n,1 a n,2 a n,j a n,p a i,j i j M n,p (K) K n p n = p M n (K) n p = 1 n n = 1 p n = p p = 1 n = 1 K M n,p (K) K 1,n 1,p K np +. K M n,p (K M n,p (K) K M n,p (K) n p

2 M n,p (K (i, j) 1, n 1, p E i,j M n,p (K) i j 1 (E i,j ) 1in M n,p (K) M n,p (K) 1jp M n,p (K) np A = (a i,j ) 1in 1jp M n,p (K) B = (b i,j ) 1ip 1jq M p,q (K) AB C = (c i,j ) 1in M n,q (K) 1jq p (i, j) 1, n 1, q, c i,j = a i,k b k,j p p AB BA ( ) ( ) BA AB A = B = k=1 (λ, µ) K 2, (A, B) M n,p (K) 2, C M p,q (K), (λa + µb)c = λac + µbc (λ, µ) K 2, A M n,p (K), (B, C) M p,q (K) 2, A(λB + µc) = λab + µac A M n,p (K), B M p,q (K), C M q,r (K), A(BC) = (AB)C (E i,j ) 1i,jn M n (K) (i, j, k, l) M n (K) 4 E i,j E k,l = δ j,k E i,l A = (a i,j ) 1in M n,p (K) A (a j,i ) 1ip M p,n (K) t A 1jp 1jn

3 (λ, µ) K 2, (A, B) M n,p (K) 2, t (λa + µb) = λ t A + µ t B A M n,p (K), t ( t A) = A (A, B) M n,p (K) 2, t (AB) = t B t A A M n,q (K) B M p,q (K) C M n,r (K) D M p,r (K) M M n+p,q+r (K) A C M = B D B ( A C D F ) ( E G ) H = BE + DF ( AE + CF AG + CH ) BG + DH A B E G C D F H B ( A C ) D t C ( t A t B ) t D

4 C i C j α C i αc i C i C i + αc j L i L j α L i αl i L i L i + αl j C i C j C i C i + C j C j C j C i C i C i + C j C j C j

5 n N i, j 1, n i j α K P i,j = E i,j + E j,i + I n E i,i E i,j D i (α) = I n + (α 1)E i,i T i,j (α) = I n + αe i,j M p,n (K) C i C j P i,j C i αc i M i (α) C j C j + αc i T i,j (α) M n,p (K) L i L j P i,j L i αl i M i (α) L i L i + αl j T i,j (α) P i,j D i (α) T i,j (α) M n (K) M n (K) n n 1 I n = (δ i,j ) 1i,jn δ i,j 1 i = j 0 M n (K (M n (K), +, ) I n n > 1 M n (K) M n (K) K K λ K (A, B) M n (K) 2 (λ.a)b = A(λ.B) = λ.(ab) 0 1 A = 0 0 B = AB = 0 A 0 B 0

6 A n N A n = A = M n (K) (I n ) A M n (K) P K[X] P(A) = 0 R n X n P A n = R n (A) 1 1 A = P 2 P(A) = 0 A n n N 2 4 M n (K) a b A = A n n N b a A = An n N M n (K) T + n (K) T n (K) n K T = (t i,j ) 1i,jn

7 T t i,j = 0 i > j T t i,j = 0 i < j T = (t i,j ) 1i,jn T t i,j = 0 i j T t i,j = 0 i j T + n (K) T n (K) n(n+1) 2 M n (K) M n (K) T + n (K) T n (K) (α 1,..., α n ) K n diag(α 1,..., α n ) n α 1,..., α n n K D n (K) D n (K) = T + n (K) T n (K) D n (K) n M n (K) A A A 1,..., A r A = A A A r

8 A M n (K) A t A = A A t A = A S n (K) A n (K) M n (K) A a j,i = a i,j (i, j) 1, n 2 A a j,i = a i,j (i, j) 1, n 2 S n (K) A n (K) M n (K) n(n+1) 2 n(n 1) 2 M n (K) S n (K) A n (K) A M n (K) A B M n (K) AB = BA = I n AB = I n BA = I n A B n K GL n (K) M n (K)

9 (A, B) GL n (K) 2 AB GL n (K) (AB) 1 = B 1 A 1 A GL n (K) A 1 ( A 1) 1 = A A GL n (K) k N A k ( A k) 1 = ( A 1 ) k A k A M n (K) A GL n (K) t A GL n (K) ( t A) 1 = t (A 1 ) A A P P(A) = 0 A 1 A A = A 3 A. A A 1. T T 1 T D D 1 D P 1 i,j = P i,j M i (α) 1 = M i ( 1 α) T i,j (α) 1 = T i,j ( α)

10 1 0 2 A = I 3 A A I 3 A I 3 A A 1 I 3 A L 2 L 2 L L 3 L 3 + 2L L 2 L 2 L 3 L 3 A 1 = L 1 L 1 2L I 3 A

11 A GL n (K) A A I n G 1,... G k G k... G 1 A = I n G k... G 1 = A 1 I n G k... G 1 I n = A 1 A A I n G 1,... G k AG 1... G k = I n G 1... G k = A 1 I n I n G 1... G k = A 1 GL n (K) GL n (K) n n n A A M n (K) B M p (K) M = A B B M 1 A C A M n (K) B M p (K) C M n,p (K) M = B A B A M n (K) A (A) A M n (K) A M n (K) ( t A) = (A) (A, B) M n (K) 2 (AB) = (BA)

12 (A, B, C) M n (K) 3 (ABC) = (CAB) = (BCA) (CBA) = (ACB) = (BAC) (ABC) = (CAB) = (BCA) (CBA) = (ACB) = (BAC) (A, B) M n,p (K) M p,n (K) (AB) = (AB) A B E K n 1 B = (e 1,..., e n ) E x E x B n B (x) x B B (x) = a 1 a 2. a n x = n a i e i i=1 1 (1, 2, 4, 3) R X 3 + 3X 2 2X 5 R 3 [X] E K n 1 B E { E Mn,1 (K) x B (x) K n n K n M n,1 (K) K n

13 E K n 1 B = (e 1,..., e n ) E F = (x 1,..., x p ) E F B n p B (F) F B B (F) = (a i,j ) 1in j 1, p, x j = 1jp n a i,j e i i=1 ((1, 2, 3), (0, 1, 2), ( 2, 1, 3), (4, 5, 1)) R ( X 3 + X 2 1, 2X 2 3, 4X 3 + 3X 2 2X 5) R 3 [X] E K n 1 B = (e 1,..., e n ) E F = (x 1,..., x n ) E F E B (F) E F K p n B 1 = (e 1,..., e p ) B 2 = (f 1,..., f n ) E F u L(E, F) u B 1 B 2 n p B1,B 2 (u) = B2 (u(e 1 ),..., u(e p )) { R f : 3 R 2 (x, y, z) (x + y z, 2x y + 3z) f B 3 B 2 ( ) R 3 R 2 f(1, 0, 0) = (1, 2) f(0, 1, 0) = (1, 1) f(0, 0, 1) = ( 1, 3) B3,B 2 (f) = 2 1 3

14 { R4 [X] R T : 5 [X] P X 4 P (3) + P(2) + P( X) T B 4 B 5 R 4 [X] R 5 [X] T(1) = 2 T(X) = 2 X T(X 2 ) = X T(X 3 ) = X 4 X T(X 4 ) = 24X 5 + X B4,B 5 (T) = E F K { p n B 1 = (e 1,..., e p ) B 2 = (f 1,..., f n ) L(E, F) Mn,p (K) E F u B1,B 2 (u) E F L(E, F) = E F A M n,p (K) A f L(K p, K n ) K p K n A { K p K A n X AX { K 3 K 2 (x, y, z) (2x 3y + 4z, x + 2z) E F K B 1 B 2 E F x E y F u L(E, F) X = B1 (x) Y = B1 (y) U = B1,B 2 (u) y = u(x) Y = UX

15 1 1 1 u L(R 3, R 2 ) U = B 3 B 2 R 3 R x = (1, 2, 3) R 3 x B 3 X = 2 0 UX = u(x) = (0, 9) 9 3 E F G K B 1 B 2 B 3 E F G u L(E, F) v L(F, G) B1,B 3 (v u) = B2,B 3 (v) B1,B 2 (u) E F K B 1 B 2 u L(E, F) u B1,B 2 (u) ( B1,B 2 (u)) 1 = B2,B 1 (u 1 ) (x 0,..., x n ) K n+1 M = (x j i ) 0i,jn M { x i Kn [X] K M n+1 P (P(x 0 ),..., P(x n )) K n [X] K n+1 E K n B E u L(E) u B n B (u) = B,B (u) E n N E E I n p ( ) E Iq 0 q,r E = p p q = p r = p 0 r,q 0 q

16 s ( ) E Iq 0 q,r E = (s E ) (s+ E ) q = (s E ) r = (s+ E ) 0 r,q I r u E F G E E B E = F G A C F u u B A B 0 B F G A u F F B A 0 F G u u B A B 0 B F G A B u F G F G B E K n B E { L(E) Mn (K) u B (u) M n (K) K n E K n B { GL(E) GLn (K) u B (u) E K n B u L(E) u B (u) B (u 1 ) = B (u) 1 { R2 [X] R 2 [X] P P(X + 1) + P(X) R 2 [X] ( (j A = i) ) ( j i) = 0 i > j A 0i,jn K n [X] A

17 (A, B) M n (K) 2 AB = I n A B AB = I n BA = I n M n (K) A B M n (C) AB = BA AB = I n + A + A 2. A M n (K) A 1 Y = AX X = t (x 1,..., x n ) Y = t (y 1,..., y n ) x 1,..., x n X = A 1 Y A 1 E K p B = (e 1,..., e p ) E φ E φ B p B (u) = B (φ(e 1 ),..., φ(e p )) K p p K p M 1,p (K) K p E K p 1 B = (e 1,..., e p ) E F = (φ 1,..., φ n ) E F B n p B (F) B (F) = (φ i (e j ) 1in 1jp A M n,p (K) A = {X M p,1 (K) AX = 0} A = {AX, X M p,1 (K)}

18 { Mp,1 (K) M n,1 (K) A A X AX A A M p,1 (K) M n,1 (K) p n K p K n A A A A M n,p (K) A M n,1 (K) A A M n,p (K) AX = 0 A A A AX = 0 X = 0 B 1 B 2 K E F n p u L(E, F) A = B1,B 2 {(u) E Mp,1 (K) u A x B1 (x) { F Mn,1 (K) u A x B2 (x) u A A A u u { R3 [X] R 3 [X] P P(X + 1) + P(X 1) 2P(X)

19 1 2 1 A = A C 2 C 2 2C 1 C 3 C 3 C 1 C 3 C 3 C A = 2, 3 A = A (A) = A

20 A M n,p (K) (A) (n, p) (A) = ( t A) A A M n,p (K) p = (A) + (A) B K E F E F = B (F) B 1 B 2 K E F u L(E, F) u = B1,B 2 (u) B K E u L(E) u = B (u) A M n (K) A A = n α K A M n,p (K) C M n,1 (K) L M 1,p (K) α L α 0 = = 1 + A 0 A C A

21 = L 2 L 2 + 3L L 3 L 3 + L 1 L 4 L L = = L 1 L = L 2 L 2 8L 1 L 3 L 3 55L = = 2 + L L = 2 + L 2 L L = 3 B B E B B B (B ) P B B B B E P B B ( P B B = B,B( E ) P B B ) 1 = P B B

22 B B E x E X = B (x) X = B (x) P = P B B X = PX X = PX X = PX E K E E F K F F u L(E, F) P = P E E Q = PF F A = E,F (u) A = E,F (u) A = Q 1 AP E K E E u L(E) P = P E E A = E (u) A = E (u) A = P 1 AP u L(E) x E y = u(x) E B, B E A, X, Y u, x, y B A, X, Y u, x, y B P B B Y = AX X = PX Y = PY Y = P 1 APX Y = A X A = P 1 AP E K E E φ E P = P E E L = E (φ) L = E (va) L = LP A A M n,p (K) A A P GL p (K) Q GL n (K) A = Q 1 AP

23 M n,p (K) 0 r (n, p) J n,p,r Ir 0 r,p r J n,p,r r 0 n r,r 0 n r,p r E F K n p u L(E, F) u r B B E F B,B (u) = J n,p,r M M n,p (K) M r M J n,p,r A = UAV = J r r = A U GL 4 (R) V GL 5 (R) M M n,p (K) t A = A A = (a i,j ) 1in M n,p (K) A 1jp (a i,j ) (i,j) I J I 1, n J 1, p

24 A B M n (K) B A P GL n (K) B = P 1 AP A B A n B n n N n Z A P B = P 1 AP B n = P 1 A n P n N n Z A 1 2 A = A A n n N 1 4

25 u E B (u) u (u) E u v K L(E) (u, v) L(E) 2 (u v) = (v u) (S) n p AX = B A M n,p (K) B M n,1 (K) X M p,1 (K) B S AX = 0 A B φ 1,..., φ p A x K n φ i (x) = b i 1 i n φ i (x) = b AX = B A AX = 0 A M n,p (K) A M p,1 (K) n (A) AX = B A M n,p (K) B M n,1 (K) B A X 0 + A X 0

26 C = (A B) B A C C = (A B ) AX = B A X = B A AX = B A n = p AX = B A A 1 B A 1 x 1 + x 2 + x x n = 1 x 1 + 2x 2 + 2x x n = 1 x 1 + 2x 2 + 3x x n = 1... x 1 + 2x 2 + 3x nx n = 1