A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa
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1 [ ] a 11 a 12 A = a 21 a 22 (1) A = a 11 a 22 a 12 a 21 (2) 3 3 n n A A = n ( 1) i+j a ij M ij i =1 n (3) j=1 M ij A i j (n 1) (n 1) A A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 A (3) i =1 A = 3 ( 1) 1+j a 1j M 1j j=1 (4) = a 11 {a 22 a 33 a 23 a 32 } a 12 {a 21 a 33 a 23 a 31 } + a 13 {a 21 a 32 a 22 a 31 } = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 11 a 23 a 32 a 12 a 21 a 33 a 13 a 22 a 31 (5) M 11 = a 22 a 33 a 23 a 32 M 12 = a 21 a 33 a 23 a 31 (6) M 13 = a 21 a 32 a 22 a 31 1
2 A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa = a n A n n A λ i (i =1,,n) A = n λ i (9) i=1 A λ i (i =1,,n) A si A = n (s λ i ) (10) (10) s =0 ii A =( 1) n A i=1 n ( λ i )=( 1) n i=1 2 n i=1 λ i (11)
3 (9) (9) 22 A 1 = 1 A adj[a] = 1 A M 11 M 21 ( 1) 1+n M n1 M 12 ( 1) 1+n M 1n M nn (12) A A = [ A 1 A 2 0 A 3 ] [, ora = A 1 0 A 2 A 3 A R (n1+n2) (n1+n2),a 1 R n1 n1,a 3 R n2 n2 A A = A 1 A 3 A 1,A 3 A [ ] A 1 A 1 1 A 1 1 A 2 A 1 3 = 0 A 1 (14) 3 ] (13) 2-3 A = (15) A 1 = (16) 3
4 2-4 (15) A A A = , A = (17) (17) (14) (16) 23 2 n x n n Q J J = x T Qx (18) J 2 x J = x T Qx 0 Q J = x T Qx > 0 Q [ J =[x 1 x 2 ] ][ x 1 x 2 ] (19) J J = x 2 1 2x 1x 2 + x 2 2 =(x 1 x 2 ) 2 (20) J x 1 = x 2 (19) [ J =[x 1 x 2 ] ][ x 1 x 2 ] (21) J J = x 2 1 2x 1x 2 +2x 2 2 =(x 1 x 2 ) 2 + x 2 2 (22) J (21) n n 4
5 Q Q n n Q i 1 i 2 i r (1 i 1 i 2 i r n) i 1 i 2 i r (1 i 1 i 2 i r n) r r Q i1,i 2 i r 2-7 Q Q = q 11 q 12 q 13 q 21 q 22 q 23 q 31 q 32 q 33 Q 1 Q 1,3 Q 1,2,3 q 11 q 12 q 13 q 11 q 12 q 13 Q(1) = q 21 q 22 q 23, Q(1, 3) = q 21 q 22 q 23 q 31 q 32 q 33 q 31 q 32 q 33 (23), Q(1, 2, 3) = q 11 q 12 q 13 q 21 q 22 q 23 q 31 q 32 q 33 Q Q 1,Q 2,Q 3,Q 1,2,Q 1,3,Q 2,3,Q 1,2,3 Q n Q(1, 2,,r) r =(1,,n) (24) 2-8 Q (24) [ Q(1) = q 11, Q(1, 2) = q 11 q 12 q 21 q 22 ], Q(1, 2, 3) = q 11 q 12 q 13 q 11 q 12 q 13 q 31 q 32 q 33 (25) Q 1 Q Q 2 Q Q 5
6 Q 1 Q Q λ i (i =1,,n) λ i 0 (i =1,,n) (26) 2 Q Q λ i (i =1,,n) λ i > 0 (i =1,,n) (27) 2-9 [ A = ] [, B = ], C = (28) (A ) A A 1 =1 A 1,2 =0 A (B ) B B 1 =1 B 1,2 = 2 B (C ) C C 1 =1 C 2 =2 C 3 =1 C 1,2 =2 C 1,3 =1 C 2,3 =3 C 1,2,3 =3 C C 3 31 dx 1 da 11 da dt dx dt =, da 1n dt dt dt = (29) dx n da n1 da nn dt dt dt t2 t2 a t2 t1 11dt a t1 1ndt Adt = (30) t1 t2 a t2 t1 n1dt a t1 nndt 6
7 d[ab] dt = d[a] dt B + Ad[B] dt, t2 t1 da t2 Bdt =[AB] t2 t1 dt t1 A db dt (31) dt 32 J(x) x J(x) x x n x = x 1 x n (32) (32) 1 n b n x J(x) J(x) =b T x x = b (33) x = (b 1 x 1 +b 2 x 2 + +b nx n) x 1 (b 1 x 1 +b 2 x 2 + +b nx n) x n 2 n n A n x J(x) =x T Ax = b (34) 1 x b 1 = Ax, =2Ax (35) b T 2 = xt A 7
8 x = xt b 1 x + bt 2 x x f = x T Ax = x, Ax = bt 1 x x + bt 2 x x = b 1 + b 2 =2Ax (36) df x = d x, Ax dx = Ax + A T x A =(a ij ) f = x, Ax = a 11 x 1 x 1 + a 12 x 1 x a 1n x 1 x n + a 21 x 2 x 1 + a 22 x 2 x a 2n x 2 x n + + a n1 x n x 1 + a n2 x n x a nn x n x n x 1,x 2,,x n f x 1 = (a 11 x 1 + a 12 x a 1n x n ) +(a 11 x 1 + a 21 x a n1 x n )) = (Ax 1 )+(A T x 1 ) f x 2 = (a 21 x 1 + a 22 x a 2n x n ) +(a 12 x 1 + a 22 x a n2 x n )) = (Ax 2 )+(A T x 2 ) f =(Ax n )+(A T x n ) x n A f x = Ax + AT x x =2Ax 8
9 f(x) f(x) = f 1 (x) f 2 (x) f n (x) f 11 x 1 + f 12 x 2 + f 1n x n = f n1 x 1 + f n2 x 2 + f nn x n (37) x f 1 (x) x f(x) 1 x = f 1 (x) x n f n(x) x 1 f n(x) x n = f 11 f n1 f 1n f nn (38) 3 n n A n x (38) A = Ax x = AT (39) a 11 a 1n, x = x 1 (40) Ax = a n1 a nn a 11 x a 1n x n x n (41) a n1 x a nn x n a 11 a n1 Ax x = a 1n a nn = A T (42) 33 x 331 n n A n b c J(x) =x T Ax + b T x + x (43) 9
10 (43) x n =1 x =0, 2 J(x) x 2 > 0 (44) x x = 1 2 A 1 b, 2 J(x) x 2 = A>0 (45) 3-1 x [ ][ ] [ 1 0 x 1 1 J(x) =[x 1 x 2 ] +[2 2] 0 1 x 2 [ ][ ] [ 2 1 x 1 2 J(x) =[x 1 x 2 ] +[2 2] 1 2 x 2 x 1 x 2 x 1 x 2 ] +2 ] +1 4 ( ) 41 n x( ) x := n x 2 i (46) 1 x 2 = x T x 2 x 0 3 x =0 x =0 4 ax = a x, a( 0) 5 x T y x y, x, y R n 6 x + y x + y, x, y R n i=1 10
11 1, 2, 3, 4 5, 6 5 y =0 y 0 2 y =0 5 y 0 1, 2 α x αy 2 =(x αy) T (x αy) =xx T 2αx T y + α 2 y T y 0 (47) α (47) α α = xt x y T y (48) (47) ( x T y ) 2 ( x T y ) 2 ( x T y ) 2 x T x 2 y T y + y T y = x T x y T y 0 ( x T y ) 2 x T xy T y (49) (49) 5 6 1, 5 x + y 2 = (x + y) T (x + y) = x T x +2x T y + y T y x 2 + y 2 +2 x T y = x 2 + y 2 +2 x T y x 2 + y 2 +2 x y = ( x + y ) 2 (50) (50) 6 42 A A := max x =1 { Ax } = { λ max [ A T A ]} 1/2 (51) 11
12 1 A > 0 A R n n,a 0 2 I =1 3 Ax A x, A R n n, x R n 4 A + B A + B 5 AB A B 1,2 3, 4, 5 3 x =0 3 x 0 Ax ( ) ( ) Ax = 1 ( x ) A x x = x 1 A x x (52) 1 x x =1 ( ) 1 A x x A (53) (53) (52) 3 4 (A + B)x, x =1 x x 0 A + B = (A + B)x 0 = Ax 0 + Bx 0 Ax 0 + Bx 0 (54) Ax 0 A, Bx 0 B (54) A + B A + B (55) 5 12
13 ABx, x =1 x x 0 AB = ABx 0 = A Bx 0 A B (56) 43 2 n n Q x 2 J(x) J(x) =x T Qx (57) λ min [Q] x 2 x T Qx λmax[q] x 2 (58) Q λ 1 λ n Q = T T ΛT, T T T = I, Λ = diag [λ 1,,λ n ] (59) T J(x) x = Tx (60) J(x) =x T Qx = x T Λ x = λ 1 x λ n x 2 n (61) λ 1 x λ 1 x 2 n λ 1 x λ n x 2 n λ n x λ n x 2 n (62) (62) λ 1 x λ 1 x 2 n = λ 1 x T x = λ min [Q]x T T T Tx = λ min [Q] x 2 (63) λ n x λ n x 2 n = λ n x T x = λmax[q]x T T T Tx = λmax[q] x 2 (64) (63) (64) (61) (62) (58) 13
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漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
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December 28, 2018
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1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
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