A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18

2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1 26 A2 Halmos Finite Dimensional Vector Spaces (1947) 27 A3 (1958) (1966) 27 A4 I,II (1966, 1969) 27 A5 (1971) 28 A6 Jordan (1976,1977) 28 A7 (1980) 28 A8 (1988, 1993) 28 A9 (1982) 29 A10 (1992) 29 1

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18 : (2007) 30 A19 (2009) 30 A20 31 B 31 B1 E I 31 C misc 31 C1 Schur 31 C2 33 D 34 D1 34 D2 35 D21 35 D22 36 D3 36 D31 36 D32 37 1 ( ) Jordan 1 U(n) := {A M(n; C); A A = A A = I} SU(n) := {A U(n); det A = 1} A B def P GL(n) st P 1 AP = B 2 n A B def P GL(n) st P T AP = B n (symmetric group) S n, S n (L A TEX \mathfrak{s} n ), Sym(n) 2

2 Jordan 21 Jordan 3 (i) ( ) (ii) ( ) (iii) 1 ( [1]) ( [2]) ( [3]) (i) ( ) Jordan (i) ([4]) 2 T [5] ( ) ( ) ( ) [4] 1 ( ) 1 ( ) Jordan ( or ) [6] ( [2] ) [7] ( ) ( ) ( [7] ) Jordan 3 Jordan (1) Jordan 3

[3] 5 4 ( [8]) ( ) 32 33 Jordan Jordan ( ) ( 1,2 ) 31 Jordan J n (a) a n Jordan ( ) a 1 0 a 1 J 1 (a) = (a), J 2 (a) =, J 3 (a) = 0 a 1, 0 a 0 0 a Jordan J m1 (a 1 ) (1) J m1 (a 1 ) J m2 (a 2 ) J mr (a r ) = J m2 (a 2 ) J mr (a r ) Jordan 31 Jordan Jordan 1, (1) m 1 = m 2 = = m r = 1 32 Jordan 32 ( Jordan ) K = R C, n N V K n T : V V (ie T n = O) V E T E Jordan T = O V T Jordan T O n n = 1 V T 1 Jordan n 2 n 1 ( Jordan ) T O, T n = O k {2, 3,, n} st T k 1 O T k = O T k 1 e 0 e V T k 1 e, T k 2 e,, T 2 e, T e, e 1 c 1 T k 1 e + c 2 T k 2 e + + c k 1 T e + c k e = 0, (c 1, c 2,, c k 1, c k K) T k 1, T k 2,, T c k = c k 1 = = c 2 = 0, c 1 = 0 4

W := span T k 1 e,, T e, e W T T (T k 1 e T k 2 e T e e) = (T k e T k 1 e T 2 e T e) = (0 T k 1 e T 2 e T e) 0 1 0 0 1 = (T k 1 e T k 2 e T e e) 1 0 0 = (T k 1 e T k 2 e T e e)j k (0) T T W : W W W T k 1 e,, T e, e J k (0) k = n W = V k < n V T U U W = {0} ( U = {0}) ( U ) V = U + W 5

V = U + W V U + W a V \ (U + W ) a T U dim U = dim U + 1, U W = {0} (U ) a U + W, T k = O T k a = 0 U + W l {1, 2,, k} st T l 1 a U + W T l a U + W u U, c 0, c 1,, c k 1 K st k 1 T l a = u + c i T i e T k 1 (T k = O ) i=0 0 = T l 1 (T k a) = T l+k 1 a = T k 1 u + c 0 T k 1 e T k 1 u = c 0 T k 1 e u U U T T k 1 u U c 0 T k 1 e W U W = {0} 0 c 0 = 0 k 1 b := T l 1 a c i T i 1 e i=1 k 1 k 1 T b = T l a c i T i e = T l a c i T i e = u i=1 k 1 b U + W ( b U + W T l 1 a = b + c i T i 1 e U + W l ) U b U dim U = dim U + 1 U T ( U T U U, T b = u U T U U U ) w U W w U v U, t K st w = v + tb tb = v + w U + W b U + W t = 0 w = v U W = {0} U W = {0} U U W = {0} V = U W T U : U U Jordan J E W T k 1 e,, T e, e E V T J k (0) J Jordan i=0 i=1 33 Jordan A = A 1 A 2 A r = A k = A k 1 A k k 2 A r 6

n Jordan J n (0) k = 0 1 0 0 0 1 J n (0) = M(n; R) 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 (1, k + 1) 1 (n k) rank J n (0) k = J n (0) k 1 = n k (k = 1, 2,, n), J n (0) n 1 O, J n (0) n = O 33 ( Jordan ) K = R C, n N V K n T : V V (ie T n = O) T Jordan Jordan T = O T O T k = O, T k 1 O (2 k n ) J Jordan (V ) T J k = O, J k 1 O J Jordan k 1 j k j J j Jordan m j (m j = 0 ) {m j } k j=1 T r i := rank T i (0 i k 1) : k k r i = rank J i = m j rank J j (0) i = m j (j i) j=0 j=i+1 7

i r k 1 = m k, r k 2 = m k 1 + 2m k, r k 3 = m k 2 + 2m k 1 + 3m k, r k j = m k j+1 + 2m k j+2 + + (j 1)m k 1 + jm k, r 1 = m 2 + 2m 3 + + (k 1)m k, r 0 = m 1 + 2m 2 + + (k 1)m k 1 + km k m k, m k 1,, m 1 (r i ) {m j } k j=1 34 Jordan ( ) 34 ( ) K = R C, n N V K n T : V V T Φ(λ) = det(λi T ) β 1,, β r, m 1,, m r f j (λ) := (λ β j ) m j, V j := ker f j (T ) = {u V ; (T β j I) m j u = 0} (j = 1,, r) V j T V = V 1 V 2 V r V j T u V j T u V j (T β j I) mj (T u) = T [(T β j I) m j u] = T 0 = 0 V = V 1 + V 2 + + V r j {1,, r} g j (λ) := Φ(λ) f j (λ) = (λ β i ) m i i j g 1 (λ),, g r (λ) 1 h 1 (λ),, h r (λ) K[λ] st (2) g 1 (λ)h 1 (λ) + + g r (λ)h r (λ) = 1 u V (2) u i := g i (T )h i (T )u u 1 + + u r = u 8

u i V i (i {1,, r}) Cayley-Hamiltion Φ(T ) = O f i (T )u i = f i (T )g i (T )h i (T )u = Φ(T )h i (T )u = 0 i j V i V j = {0} f i (λ) f j (λ) φ i (λ), φ j (λ) K[λ] st φ i (λ)f i (λ) + φ j (λ)f j (λ) = 1 u V φ i (T )f i (T )u + φ j (T )f j (T )u = u u V i V j f i (T )u = f j (T )u = 0 u = 0 V i V j = {0} 35 ( Jordan ) n N V C n T : V V V E T E Jordan T β 1,, β r m 1,, m r V i := ker(β i I T ) m i V i T V = V 1 V r T β i I V i N i := (T β i I) Vi V i E i N i E i Jordan Jordan J i T Vi J i β i I mi (I mi m i ) E 1,, E r V E E T (J 1 β 1 I m1 ) (J r β r I mr ) Jordan 35 ( ) 36 (1 Jordan ) K = R C, n N V K n T : V V α K (n ) V E T E Jordan Jordan Jordan ( ) S := T αi T Φ T (λ) = (λ α) n S Φ S (λ) = λ n Hamilton-Cayley S n = Φ S (S) = O 32 V E S E 0 Jordan Jordan J d1 (0) J dr (0) 9

T = S + αi E J d1 (α) J dr (α) Jordan ( ) V 2 E, E T Jordan J, J J αi n, J αi n E, E T αi Jordan T αi J αi n, J αi n ( ) 0 k, l N, t 1,, t k, s 1,, s l 0 st J αi n = J t1 (0) J tk (0), J αi n = J s1 (0) J sl (0) J = J t1 (α) J tk (α), J = J s1 (α) J sl (α) 33 j m j k = l t 1,, t k s 1,, s k 37 n N V C n T : V V T Jordan Jordan V E T Jordan J J T ( ) T α 1,, α r, m 1,, m r α i T ker(α i I T ) m i W i J α i Jordan E ( ) : e i,1, e i,2,, e i,mi ( m i ) W i = span e i,1, e i,2,, e i,mi W i span e i,1, e i,2,, e i,mi 1 m i W i T E i := e i,1, e i,2,, e i,mi W i T Wi E i J i J ( α i ) Jordan : J i = J k1 (α i ) J k2 (α i ) J kni (α i ) 36 J i k E T α i 4 Jordan (2) ( ) 1 ( ) J 1 J k (α i ) E e l,,e l+k 1 T e l = α i e l, T e l+1 = e l + α i e l+1,, T e l+k 1 = e l+k 2 + α i e l+k 1 e l,, e l+k 1 W i 10

41 x x ( x ) ( ) x 2 + 3x 1 2x A(x) = x 2009 75x 3 34x 56 K = R K = C K[x] m n M(m, n; K[x]) m = n M(n; K[x]) K K[x] M(m, n; K) M(m, n; K[x]) A(x) M(m, n; K[x]) A(x) 2009 A(x) deg A(x) K x x (M(m, n; K[x]) M(m, n; K)[x]) ( ) ( ) ( ) ( ) x 2 + 2x + 3 4x + 5 1 0 = x 2 2 4 3 5 + x + 6x 2 + 7x 8 9 6 0 7 0 8 9 n N A(x) M(n; K[x]) (3) A(x)B(x) = B(x)A(x) = I B(x) M(n; K[x]) (I n ) A(x) A(x) (3) B(x) A(x) A(x) 1 A(x) det A(x) K, det A(x) K, det A(x) 0 A(x) A(x) A(x) 1 41 K = R K = C n N A(x), B(x) M(n; K[x]) A(x) = A 0 x k + A 1 x k 1 + + A k 1 x + A k, A 0 O, B(x) = B 0 x l + B 1 x l 1 + + B l 1 x + B l, B 0 GL(n; K) A(x) = B(x)Q(x) + R(x), deg R(x) < deg B(x) x Q(x) R(x) ( A(x) = Q(x)B(x) + R(x) Q(x), R(x) ) ( A(x) = O B(x) B 0 GL(n; K) ) 11

42 x m, n N A, B M(m, n; K) A B def Q GL(m; K), P GL(n; K) st A = QBP M(m, n; K) E r = diag(1, 1,, 1, 0,, 0) ( 1 0 ) A E r rank A = r A E r A B (A B B A ) x 42 K = R K = C, m, n N A(x), B(x) M(m, n; K[x]) A(x) B(x) A(x) = Q(x)B(x)P (x) Q(x) M(m; K[x]), P (x) M(n; K[x]) A(x) B(x) A(x) B(x) 2 M(m, n; K[x]) 12

43 ( ) n (elementary matrix) 3 1) ( (interchange matrix)) i j 1 1 0 1 1 P n (i, j) := 1 1 0 1 1 ( i j ) 2) c K (ie, c K, c 0) Q n (i; c) := i diag(1 1 1 c 1 1) = 1 1 c 1 1 ( (i, i) c ) 3) i j, c(x) K[x] R n (i, j; c(x)) := I + c(x)e ij ( E ij ) ( (i, j) c(x) ) 44 ( ) ( ) P n (i, j) = E ij + E ji + k i,j E kk = I E ii E jj + E ij + E ji (i j), Q n (i; c) = I + (c 1)E ii, R n (i, j; c(x)) = I + c(x)e ij (i j) (I n ) 13

45 ( ) A(x) M(m, n; K[x]) (1) A(x) P m (i, j) A(x) i j (2) A(x) P n (i, j) A(x) i j (3) A(x) Q m (i; c) A(x) i c (4) A(x) Q n (i; c) A(x) i c (5) A(x) R m (i, j; c(x)) A(x) i j c(x) (6) A(x) R n (i, j; c(x)) A(x) i j c(x) 46 ( ) 45 (elementary transformation) (, elementary row operation) (, elementary column operation) 47 ( ) P n (i, j) 1 = P n (i, j) (i j), Q n (i; c) 1 = Q n (i; c 1 ) (c 0), R n (i, j; c(x)) 1 = R n (i, j; c(x)) (i j) 48 A(x) B(x) B(x) A(x) 49 A(x) B(x) A(x) B(x) 410 ( ) n N, A(x) M(n; K[x]) k {1,, n} A(x) k ( 1 ) A(x) k d k (x) k 0 d k (x) = 0 411 A(x) k d k (x) (1) i j A(x) P n (i, j) k 14

d k (x) (2) c 0 A(x) Q n (i; c) k c d k (x) (3) i j, c(x) K[x] A(x) R n (i, j; c(x)) A(x) i j c(x) Ã(x) Ã(x) k d k (x) (a) A(x) k i Ã(x) (b) A(x) k i j Ã(x) ( ) (c) A(x) k i j Ã(x) det a i (x) + c(x)a j (x) = det a i (x) + c(x) det a j (x) 1 2 det A(x) k d k (x) d k (x) (a), (b), (c) d k (x) d k (x) d k (x) d k (x) d k (x) = d k (x) 15

43 412 (x ) K = R K = C, n N A(x) M(n; K[x]) r {0, 1,, n}, e 1 (x),, e r (x) K[x] st j {1,, r} e j (x) 1 e j (x) e j+1 (x) (j = 1, 2,, r 1) A(x) e 1 (x) e 2 (x) (4) e r (x) 0 = diag(e 1 (x), e 2 (x),, e r (x), 0,, 0) 0 r, e 1 (x),, e r (x) A(x) d k (x) := A(x) k 1 (5) e 1 (x) = d 1 (x), e j (x) = d j(x) d j 1 (x) (j = 2,, r) r A(x) e 1 (x),, e r (x) A(x) (elementary divisor) (4) A(x) ( ) (4) d 1 (x) = e 1 (x), d 2 (x) = e 1 (x)e 2 (x), d r (x) = e 1 (x)e 2 (x) e r (x), d r+1 (x) = = d n (x) = 0 (5) r, e 1 (x),, e r (x) A(x) ( ) n n = 1 ( ) n > 1 n 1 x A(x) M(n; K[x]) A(x) = O (4) A(x) O A(x) (1, 1) 0 ( ) 1 (1, 1) e 1 (x) b 12 (x) b 1n (x) b 21 (x) b 22 (x) b 2n (x) B(x) = b n1 (x) b n2 (x) b nn (x) 16

B(x) A(x) B(x) 1, 1 e 1 (x) e 1 (x) e 1 (x) 0 0 0 b22 (x) b2n (x) B(x) = 0 bn2 (x) bnn (x) e 1 (x) b ij (x) ( ) B(x) = e 2 (x) b22 (x) b2n (x) bn2 (x) bnn (x) e r (x) e j (x) (j = 2,, r) 1 e 2 (x) e 3 (x) e r (x) e 2 (x) B(x) 1 e 1 (x) e 2 (x) A(x) e 1 (x) e 2 (x) e r (x) 0 0 413 ( ) K = R K = C, n N, A(x) M(n; K[x]) (1) A(x) I ( ) (2) A(x) A(x) 0 0 (3) A(x) A(x) ( ) (1) 17

A(x) det A(x) 0 0 A(x) I det A(x) det I = 1 0 c K st det A(x) = c A(x) A(x) c K st det A(x) = c A(x) 0 K n e 1 (x) = e 2 (x) = = e n (x) = 1 A(x) (2) A(x) (1) P 1 (x),, P k (x), Q 1 (x),, Q l (x) Q 1 (x) Q l (x)a(x)p 1 (x) P k (x) = I A(x) = Q l (x) 1 Q 1 (x) 1 P k (x) 1 P 1 (x) 1 A(x) (3) A(x) (2) A(x)P 1 (x) P k (x)q 1 (x) Q l (x) = I, P 1 (x) P k (x)q 1 (x) Q l (x)a(x) = I A(x) I 414 K = R K = C, n N A(x), B(x) M(n; K[x]) 3 (i) A(x) B(x) P (x), Q(x) M(n; K[x]) B(x) = Q(x)A(x)P (x) (ii) A(x) B(x) (iii) A(x) B(x) ( ) (i) = (ii) (ii) = (i) (iii) = (ii) A(x) B(x) ( ) (ii) = (iii) 415 ( ) K = R K = C, n N A, B M(n; K) P GL(n; C) st B = P 1 AP xi A xi B ( ) B = P 1 AP xi B = xi P 1 AP = P 1 (xi A)P P xi A xi B 18

( ) xi A xi B P (x) Q(x) (xi A)P (x) = Q(x)(xI B) P 1 (x), Q 1 (x), P, Q st P (x) = P 1 (x)(xi B) + P, Q(x) = (xi A)Q 1 (x) + Q (xi A)(P 1 (x)(xi B) + P ) = ((xi A)Q 1 (x) + Q)(xI B) (xi A)(P 1 (x) Q 1 (x))(xi B) = x(q P ) + AP QB P 1 (x) = Q 1 (x), P = Q, AP = QB 416 A M(n; C) xi A e 1 (x),, e n (x) e n (x) A e 1 (x) e n (x) A det(xi A) 44 441 2 ( ) 1 ( ) ( ) 21 ( ) 2 19

442 Jordan A xi A ( Jordan ) xi A (i) e 1 (x) e 2 (x) e n (x) (ii) e j (x) 1 (j = 1,, n) e 1 (x),, e n (x) xi A e 1 (x),, e n (x) xi A d k (x) (k = 1,, n) ( ) ( d n (x) = det(xi A) ) ( ) α 0 (1) A = e 1 (x) = x α, e 2 (x) = x α 0 α xi A = ( ) x α 0 0 x α ( (x α) (x α)) d 1 (x) = x α, d 2 (x) = (x α) 2 ( ) α 0 (2) A = ( α β) e 1 (x) = 1, e 2 (x) = (x α)(x β) 0 β xi A = ( ) x α 0 0 x β GCD(x α, x β) = 1 ( ) ( ) ( ) x α 0 x α x β β α x β 1 x β β x 0 x β 0 x β β x x β β α x β 1 0 β x β x = 1 0 β x (x α)(x β) x β (x β) β α β α β α β α 1 0 ( ) (x α)(x β) 1 0 0 0 (x α)(x β) β α ( 1 (x α)(x β)) d 1 (x) = GCD(x α, x β) = 1, d 2 (x) = det(xi A) = (x α)(x β) 20

( ) α 1 (3) A = e 1 (x) = 1, e 2 (x) = (x α) 2 0 α ( ) ( ) ( ) xi A = x α 1 1 x α 1 x α 0 x α x α 0 (x α) 0 ( ) ( ) 1 0 1 0 (x α) (x α) 2 0 (x α) 2 d 1 (x) = 1, d 2 (x) = (x α) 2 α 0 0 (4) A = 0 α 0 e 1 (x) = x α, e 2 (x) = x α, e 3 (x) = x α 0 0 α x α 0 0 xi A = 0 x α 0 0 0 x α ( (x α) (x α) (x α)) d 1 (x) = x α, d 2 (x) = (x α) 2, d 3 (x) = det(xi A) = (x α) 3 α 1 0 (5) A = 0 α 0 e 1 (x) = 1, e 2 (x) = x α, e 3 (x) = (x α) 2 0 0 α x α 1 0 1 x α 0 1 x α 0 xi A = 0 x α 0 x α 0 0 α x 0 0 0 0 x α 0 0 x α 0 0 x α 1 x α 0 α x 0 0 α x (x α) 2 0 0 (x α) 2 0 0 0 x α 0 0 x α 0 0 x α 0 0 x α 0 x α 0 0 (x α) 2 0 0 0 (x α) 2 (1 (x α) (x α) 2 ) d 1 (x) = 1, d 3 (x) = det(xi A) = (x α) 3 1 0 = (x α) 0 x α 2 d 2 (x) = x α e 1 (x) = d 1 (x) = 1, e 2 (x) = d 2(x) d 1 (x) = x α 1 = x α, e e (x) = d 3(x) d 2 (x) = (x α)3 x α = (x α)2 21

α 1 0 (6) A = 0 α 1 e 1 (x) = 1, e 2 (x) = 1, e 3 (x) = (x α) 3 0 0 α x α 1 0 1 x α 0 1 x α 0 xi A = 0 x α 1 x α 0 1 α x 0 1 0 0 x α 0 0 x α 0 0 x α 1 x α 0 α x 0 1 α x (x α) 2 1 0 (x α) 2 1 0 0 x α 0 0 x α 0 0 x α 0 1 (x α) 2 0 1 (x α) 2 0 x α 0 0 α x 0 0 1 0 = 0 1 0 0 (x α) 2 (α x)(x α) 2 0 (x α) 2 (x α) 3 0 1 0 0 0 (x α) 3 (1 1 (x α) 3 ) d 1 (x) = 1, d 3 (x) = det(xi A) = (x α) 3 2 1 0 x α 1 = 1 d 2 (x) = 1 e 1 (x) = d 1 (x) = 1, e 2 (x) = d 2(x) d 1 (x) = 1 1 = 1, e 3(x) = d 3(x) d 2 (x) = (x α)3 1 = (x α) 3 α 0 0 (7) A = 0 α 0 ( α β) e 1 (x) = 1, e 2 (x) = x α, e 3 (x) = (x α)(x β) 0 0 β 22

x α 0 0 x α 0 x β xi A = 0 x α 0 0 x α 0 0 0 x β 0 0 x β β α 0 x β 1 0 x β 0 x α 0 0 x α 0 (x β) 0 x β x β 0 x β β α 1 0 x β 0 x α 0 x β 0 x β + (x β) x β = 0 x α 0 x β (x α)(x β) 0 β α β α β α β α 0 x α 0 (x α)(x β) 0 x α 0 0 0 0 0 (x α)(x β) β α GCD(x α, x β) = 1 d 1 (x) = 1, d 3 (x) = det(xi A) = (x α) 2 (x β) 0 2 (x α) 2 (x α)(x β) d 2 (x) = x α e 1 (x) = d 1 (x) = 1, e 2 (x) = d 2(x) d 1 (x) = x α, e 3(x) = d 3(x) d 2 (x) = (x α)2 (x β) x α = (x α)(x β) α 1 0 (8) A = 0 α 0 ( α β) e 1 (x) = 1, e 2 (x) = 1, e 3 (x) = (x α) 2 (x β) 0 0 β x α 1 0 1 x α 0 1 x α 0 xi A = 0 x α 0 x α 0 0 α x 0 0 0 0 x β 0 0 x β 0 0 x β α x (x α)(α x) 0 = α x (x α) 2 0 0 0 x β 0 0 x β 0 (x α) 2 0 = 0 (x α) 2 0 0 0 x β 0 0 x β 0 0 x β 0 x β 0 0 (x α) 2 0 0 0 (x α) 2 (β α) 2 = [ x + (2α β)] (x β) + (x α) 2 23

2 x + (2α β) 3 3 2 0 x β 0 0 (β α) 2 (x α) 2 0 (β α) 2 (x α) 2 0 x β 0 0 (β α) 2 (x α) 2 (x α)2 (β α)2 (β α) 2 (x α)2 = 0 (β α) 2 0 (x β)(x α)2 0 x β 0 x β 0 (x β) (β α) 2 (β α) 2 0 (β α) 2 0 (x β)(x α)2 0 1 0 0 0 0 0 (x α) 2 (x β) (β α) 2 d 1 (x) = 1, d 3 (x) = det(xi A) = (x α) 2 (x β) 0 2 (x α) 2 1 0, (x α)(x β), = (x β) 0 x β d 2 (x)=1 e 1 (x) = d 1 (x) = 1, e 2 (x) = d 2(x) d 1 (x) = 1 1 = 1, e 3(x) = d 3(x) d 2 (x) = (x α)2 (x β) 1 = (x α) 2 (x β) α 0 0 (9) A = 0 β 0 ( α β, β γ, γ α) e 1 (x) = 1, e 2 (x) = 1, e 3 (x) = 0 0 γ (x α)(x β)(x γ) x α 0 0 xi A = 0 x β 0 0 0 x γ d 1 (x) = GCD(x α, x β, x γ) = 1, d 2 (x) = GCD((x α)(x β), (x β)(x γ), (x γ)(x α)) = 1, d 3 (x) = det(xi A) = (x α)(x β)(x γ) e 1 (x) = d 1 (x) = 1, e 2 (x) = d 2(x) d 1 (x) = 1 1 = 1, e 3 (x) = d 3(x) d 2 (x) = (x α)(x β)(x γ) 1 = (x α)(x β)(x γ) 24

443 [9] A(x) = (a ij (x)) x n M(n; R) ( R := C[x]) (1) a ij (x) 0 a 11 (x) (2) j = 2, 3,, n a 1j (x) a 11 (x) : a 1j (x) = q 1j (x)a 11 (x) + r 1j (x), det r 1j (x) < deg a 11 (x) 1 q 1j (x) j (3) a 1j (x) (j = 2,, n) 0 a 11 (x) (2) a 1j (x) = 0 (j = 2,, n) (4) a 1j (x) = a j1 (x) = 0 (j = 2,, n) : a 11 (x) 0 0 0 A(x) = A (x) 0 a 11 (x) a ij (x) (i, j = 1,, n) a 11 (x) 417 n N, A(x) M(n; C[x]) A(x) det A(x) C \ {0} A(x)B(x) = B(x)A(x) = I n B(x) M(n; C[x]]) (elementary divisor) 45 25

0 2 3 418 ( p ) A = 3 1 3 xi A 3 2 6 x 2 3 2 x 3 1 x 3 xi A = 3 x 1 3 C 2 C 3 x 1 3 3 x 1 3 3 2 3 2 x 6 2 3 x 6 1 3 x 6 x 1 3 x x 1 3 + 3 x 1 2 2 2 = x 1 (x 1)(x + 2) 3(x 3) 2 2 2 1 3 x 1 x 6 + 3 1 1 3 x x 3 (x 1)(x + 2) 3(x 3) 0 2 2 0 3 x x 3 (x 3)(x + 2) 3(x 3) 0 3 x x 3 0 2 2 0 3 x 0 (x 3)(x + 2) 3(x 3) (x 3)(x + 2) 0 2 2 2 = 0 3 x 0 (x 3)(x + 2) (x 1)(x 3) 0 2 2 0 3 x 0 (x 1)(x 3) 0 x 3 0 0 0 0 0 (x 1)(x 3) 2 e 1 (x) = 1, e 2 (x) = x 3, e 3 (x) = (x 1)(x 3) 5 Jordan [2], [10], [6], Strang [11], Moscow University Vestnik 26 Filippov, [12] A A1 [7] [10] 3 2 [1], 26

A2 Halmos Finite Dimensional Vector Spaces (1947) Halmos [13] A3 (1958) (1966) 1978 [1] [4] [1] ( ) ( [14] ) ( ) ([4] ) ( ) [4] ( ) Gauss LDU 3 [4] [1] (3 ) 2 2 ( ) 3 4 Jordan [4] [3] [3] 5 [1], [4] [15], [8] A4 I,II (1966, 1969) [16], [17] 3 ( 2 ) 4 5 [3] [4] 27

A5 (1971) [10] ( ) A6 Jordan (1976,1977) [2] ( ) Jordan Jordan A7 (1980) [18] [19] ( ) ( ) A8 (1988, 1993) ( 1 ) [20] 1 (CG ) 1 28

( ) ( ) 6 2 10 ( 2 ) 2 = 2, 2 > 0 ( 2 ) ( ) A9 (1982) [6] Jordan ( A A + εf (F ε R) Jordan ) Kato [21] [22] ( ) A10 (1992) [9] ( ) A11 (1993,1994) [23] A12 (1994) [24] A13 Trefethen and Bau Numerical Linear Algebra (1997) Trefethen and Bau[25] 6 29

A14 (1999) [26] A15 (2003) [27] A16 (2004) [28] A17 (2007) [29] A18 : (2007) [30] A19 (2009) [31] ( ) Version 2 ( ) 30

A20 [32] BASIC [33] [34] [35] ( ) [36] B B1 E I E [7], [4], [3], [1],[14], [26], [30], [16], [17], [37], [38] I [39] ( I E) I [2], [20], [24], [10], [31], [29], [20], [6], Strang [11], [28], [23], [40], [9] C misc C1 Schur [1] IV 3 C1 (Schur ) A M(n; C) U S(n) λ 1 U * AU = ( ) 0 λ n ( : [1] Schur ) n λ 1 A 1 u λ 1 u 1 := 1 u u u 1,, u n Q := (u 1 u n ) 31

v C n 1, A M(n 1; C) st ( Q AQ = λ 1 v 0 A Q O(n 1) st λ 2 Q A Q * = 0 λ n ( ) 1 0 T U := Q U O(n) 0 Q ( ) ( ) U 1 0 T AU = Q 1 0 T AQ = 0 Q 0 Q ( ) ( ) λ 1 v 1 0 T = = 0 Q A 0 Q λ 1 * = 0 λ n ) ) ( ) ( ) λ 1 v 1 0 T 0 Q ) 0 A 0 Q ( 1 0 T ( λ 1 v Q 0 Q A Q Hermite A W A (W := Span u 1 ) 1 ( v v Q ) C2 A M(n; R) λ 1 Q O(n) st Q T * AQ = 0 λ n C3 ( ) A n (AA = A A) U O(n), λ 1,, λ n C st U AU = diag(λ 1,, λ n ) U O(n) st U AU = λ 1 * 0 λ n C = (c ij ) A = UCU A A = AA UC CU = UCC U C C = CC i {1,, n} c ki c ki = c ik c ik i k n 1 k i 32

c ii 2 i = 1 i = 2 i = 3 i<k n c ik 2 = 1 k<i c ki 2 n c 1k 2 = 0 c 12 = c 13 = = c 1n = 0 k=2 n c 2k 2 = c 12 2 = 0 c 23 = c 24 = = c 2n = 0 k=3 n c 3k 2 = c 13 2 + c 23 2 = 0 c 34 = c 35 = = c 3n = 0 k=4 C 0 C C2 C4 ( ) A M(m, n; C) r := rank(a A), A A 0 r1, 2, rr 2 (r i > 0) U 1 U(m), U 2 U(n) st r 1 0 r r U 1 AU 2 = 0 0 0 m = n B := A A Hermite ( B = (A A) = A (A ) = A A = B) ( (Bx, x) = (A Ax, x) = (Ax, Ax) 0) U U(n) st β 1 U 0 BU =, β i 0 0 β n B β i 0 U β 1 = = β r > 0, β r+1 = = β n = 0 (0 r n) AU (t 1 t 1 t n t n ) (t 1 t n ) = (AU) (AU) = U A AU = U BU = β 1 0 0 β n = β 1 0 0 β r 0 0 0 (t i, t j ) = β i δ ij (1 i n, 1 j n) 33

1 r 1 t 1,, 1 r r t r t i = 0 (r + 1 i n) u 1 := 1 r 1 t 1,, u r := 1 r r t r u 1,, u n C n u i (r + 1 i n) t j (1 j r) u 1 u r u r+1 u n (t 1 t n ) = r 1 0 0 0 r r 0 0 ( 2 0 t i = 0 (r + 1 i n) 0 (u i, t j ) = 0 (r + 1 i n, 1 j r) ) T 2 := (u 1 u n ), T 1 := U u 1 u r u r+1 (t 1 t n ) = T2 AU = T2 AT 1 u n T a 2 stat 1 = Σ A = T 2 ΣT 1 A (singular value decomposition, SVD) [1] III 3 Schmidt (p 101) QR ( ( ) ) IV 4 (p 162) Cholesky (, QR, Cholesky, Schur, ) ( ) LU 1 ( ) D D1 V K V V V := {f; f : V K } f, g V, α K f + g, αf (f + g)(x) = f(x) + g(x), 34

(αf)(x) = α f(x) (e 1,, e n ) V i f i : V K f i ( n j=1 c j e j ) = c i f i V (f 1,, f n ) V (e 1,, e n ) D1 V K n V K n V V (V ) V a V ϕ a : V K, ϕ a (f) = f(a) f V ϕ a ϕ a (V ) D2 V K n V α ϕ a V D2 D21 V W V e 1,, e n W f 1,, f m n m c ij e i f j i=1 j=1 ( e i f j 1 ) V W 35

D3 V W K (1) (T1), (T1 ), (T2) T Φ: V W T (T1) x 1,, x r V 1 y 1,, y r W r Φ(x i, y i ) = 0 y i = 0 (1 i r) r=1 (T1 ) y 1,, y r W 1 x 1,, x r V r Φ(x i, y i ) = 0 x i = 0 (1 i r) r=1 (T2) T Φ(x, y) (x V, w W ) (2) T Φ : V W T Φ (x, y) = ρ(φ(x, y)) (x V, y W ) ϕ: T T (6) T = V W, Φ(x, y) = x y D22 (7) L (V, W ) W V D3 D31 V 2 T T = V 1 V r, V i = V V r T T r V i = V i p V i = V i q T p q r i V i = V T r {}}{ V V 36

r i V i = V T r {}}{ V V dim V = n, T r dim T = n r D32 r N S r r r T r = T r (V ) := σ S n P σ : T r T r r {}}{ V V P σ (e i1 e ir ) = e σ(i1 ) e σ(ir) (i 1,, i r {1,, n}) x 1,, x r V P σ (x 1 x r ) = x σ(1) x σ(r) P σ V P σ P τ = P τσ τσ := τ σ r t σ S r P σ (t) = t r t σ S r P σ (t) = sign σ t S := σ S r P σ, A := σ S r sign σ P σ [1], (1958, 1974), [2], Jordan, (1990),, Jordan I, II, (1976,1977) [3], (1985) 37

[4], (1966) [5], (1987) [6],, (1982) [7], (1957) [8],, pp 12 13 (2000 6 ),,, (2007) ( UP 2000 11 ) [9], (1992, 1992, 1994) [10], (1971, 2002) [11] Strang, G S: Linear algebra and its applications, Academic Press (1976) [12] Jordan,, 55 4 2003 10, pp 424 429 (2003) [13] Halmos, P R: Finite Dimensional Vector Spaces, Princeton University Press (1947) [14], (1997) [15], 2000 6, pp 10 11 (2000) [16] I, (1966) [17] II, (1969) [18], (1980) [19], (1980) [20] Chatelin, F: Valeurs propres de matrices, Masson, Paris (1988), ( ) F,,,, (1993) [21] Kato, T: Perturbation Theory for Linear Oprators, Springer Verlag (1966) [22], (1999) [23], (2003),, I, II,, (1993, 1994) [24],,,, (1994) [25] Trefethen, L N and Bau III, D: Numerical Linear Algebra, SIAM (1997) [26], (1999) [27], (2003) 38

[28],, (2004) [29] David A,, (2007),, [30] :, (2007) [31],,,, (2009) [32] BASIC, (1985) [33], (1971) [34], (1977) [35], (1993) [36], (1995) [37], (2002) [38], (1992) [39] 30, (1991) [40], (2000) 39