+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm.....

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+ http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm (1) n m () (n, m) ( ) n m B = ( ) 3 2 4 1 (2) 2 2 ( ) (2, 2) ( ) C = ( 46 2 51 ) 299 31 26 102 991 (3) 2 4 ( ) ( ) ( ) 1

(= ) i j (i, j) ( ) A (i, j) A ij B 11 = 3, C 23 = 102 (4) (1) A ij = a ij (5) A (i, j) a, b, c x, y A (i, j) a ij B, C A ij = a ij A ij A (i, j) ( ) ( ) (n, m) A (k, l) D 1,2 1 A D ( ) n = k, m = l (6) 2 A D i = 1, 2,, n, j = 1, 2,, m A ij = D ij (7) A D A = D 1 B, C (4) 2 * 1 5 3 2 a, b, c, d, e, f ( ) a + 5 a + 2b 7 4 0 5c 8d 6, ( 3 b + 13 2e ) e f 5 3f + 2 URL http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf *1 Hint: 2

2 ( ) 21 (n, m) E a ae 11 ae 1j ae 1m ae ae i1 ae ij ae 1m ae n1 ae nj ae nm (8) a (8) := (8) E a ae (2) B 3 3B = ( ) 3 3 3 ( 2) = 3 4 3 1 ( 9 ) 6 12 3 1 (9) 4 (3) C 15 15C 22 ( ) A, F A F ( ) A F (F (n, m) ) A 11 + F 11 A 1j + F 1j A 1m + F 1m A + F A i1 + F i1 A ij + F ij A im + F im A n1 + F n1 A nj + F nj A nm + F nm (10) A + F ( ) 6 11 G = 9 8 (11) B B + G = ( ) ( ) 3 + ( 6) ( 2) + 11 3 9 = 4 + 9 1 + ( 8) 13 7 (12) B C ( ) 76 4 51 299 B + C = 09 16 102 991 3

B + C = ( 46 2 21 ) 099 31 26 142 981 ( ) 5 6 7 ( ) 7 9i 1 86 2 ( ), 1 1, 1 2 3 4 5 6 7 8 9 10, 2 7 25i 11 12 13 14 15 ( ) 1 2i 4 8i, 1 1 2 3 5 ( ) 8 13 21 34 55 2 + 14i 314 8i, 031 1324 2 5 89 144 233 377 610 G + B B + G G + B 6 (1) 6 (2) A F * 2*3 8 (1) 2 2 7 0 1 + 6 1 4 10 + 1 8 9 3, 3 4 4 5 7 1 2 7 0 1 + 6 1 4 10 + 1 8 9 3 3 4 4 5 7 1 (2) (1) (3) A E, F 23 O () A A + O = O + A = A (13) O ( ) O (10) O A (n, m) (10) O 0 A + A = A + A = O (14) *2 (8) (12) *3 (i, j) 4

(n, m) A A A + ( A) = ( A) + A = O (15) A 1 (10) A 11 A 1j A 1m ( A) = A i1 A ij A im A n1 A nj A nm (16) ( A) ij = A ij (17) ( A) (i, j) A (i, j) -1 (17) a + ( b) = a b A 11 F 11 A 1j F 1j A 1m F 1m A + ( F ) = A F = A i1 F i1 A ij F ij A im F im A n1 F n1 A nj F nj A nm F nm (18) A + ( B) A B 9 10 (10) O 0 5 24 ( ) A, H n m m l m m m A 1k H k1 A 1k H kj A 1k H kl m m m AH A jk H k1 A ik H kj A ik H kl m m m A nk H k1 A nk H kj A nk H kl (19) 5

AH (i, j) (AH) ij m (AH) ij = A ik H kj = A i1 H 1j + A i2 H 2j + + A im H mj (20) n l AH HA F A n l BC CB (BC) 11 = B 11 C 11 + B 12 C 21 = 20 (21) 11 12 BC BC (1) ( ) 2 1 1 1 2 3 1 2 3 2 1 0 4 0 3 (3) ( ) ( ) 1 i 1 + i 0 i 1 0 1 i (2) 2 1 0 1 1 2 1 2 0 3 2 3 (4) ( ) 2 2 1 i 3i 1 + i 3i 13 A 2 O ( ) AO = O, OA = O 14 (1) 2 ( ) 1 1 1 ( ) 1 2 3 2 2 1 1 1 1 2 3 2 1 2 3, 1 2 3 2 2 1 0 2 1 0 4 0 3 3 4 0 3 3 (2) (1) (3) n m A m l H,k n J 15 (1) 2 ( ) 2 1 1 1 2 3 + 2 1 0, 2 1 0 1 2 3 4 0 3 6 1 2 1 0 3 2

( ) 2 1 1 ( ) 1 2 3 1 2 3 1 2 3 + 2 1 0 1 2 1 2 1 0 2 1 0 4 0 3 0 3 2 (2) 2 2 1 1 1 2 3 + 2 1 0 1 2 1 1 2, 4 0 3 0 3 2 3 2 1 1 1 2 3 1 2 + 2 1 0 1 2 1 1 2 4 0 3 3 0 3 2 3 (3) (1) (2) (4) (3) n m A, E m l H,k n J 16 6-8,13-15 25 T 251 T (Transpose) n m A m n a 11 a n1 A T (22) a 1m a nm ( ) A T (i, j) (A T ) ij A ji (23) A T (i, j) A (j, i) B T 2 2 (B T ) 12 = B 21 = 4 (24) t t A (25) 17 B T C T 7

252 (Complex Conjugate) Z Z Z Z11 Z1j Z 1m Z Zi1 Zij Zim Zn1 Znj Znm (26) Z Z (Z ) ij (Z ij ) (27) Z (i, j) Z (i, j) ( ) i 1 + i Z = 2 i 1 + 3i (28) Z = ( i ) 1 i 2 + i 1 3i (29) Z (30) 253 (Hermite Conjugate) ( ) Z11 Zn1 Z (31) Z1m Znm (Z ) ij (Z ji ) (32) Z (i, j) Z (j, i) (28) (Z ) 11 = (Z 11 ) = i, (Z ) 12 = (Z 21 ) = 2 + i (33) Z (34) (adjoint) 8

1 18 (1) ( 1 + i ) 2 + 3i 0 1 7i (2) 2 1 + i 3i (3) i 0 1 3 1 2 + i 2 1 + i 2 1 2i 2 (4) ( ) 3 1 0 1 7 4 19 *4 X, Y a (X T ) T = X, (ax) T = ax T, (X + Y ) T = X T + Y T, (XY ) T = Y T X T (35) (X ) = X, (ax) = a X, (X + Y ) = X + Y, (XY ) = X Y (36) (X T ) = (X ) T (37) 20 (35) (37) X, Y a (X ) = X, (ax) = a X, (X + Y ) = X + Y, (XY ) = Y X (38) 3 n n 1 n 1 n n a 1 b 1 a 2 a =, b 2 b = b n 1 (39) a n b n n a b = a 1 b 1 + a 2 b 2 + + a n b n = a i b i (40) i=1 a a T = ( a 1 a 2 a n ) (41) *4 9

a T 1 n a b = a T b = ( ) b 2 a 1 a 2 a n b 1 (42) 1 n n 1 1 1 m n m v i (i = 1, 2,, n) n (v 1 ) 1 (v j ) 1 (v n ) 1 ( ) v1 v 2 v n = (v 1 ) i (v j ) i (v n ) i (v 1 ) m (v j ) m (v n ) m b n (43) n u i (i = 1, 2,, m) m (u 1 ) 1 (u 1 ) j (u 1 ) n u 1 u 2 = (u i ) 1 (u i ) j (u i ) n u m (u m ) 1 (u m ) j (u m ) n (44) (19) AH m n A m l H nl 4 ( ) 41 ( ) n n n () n K, L KL LK KL LK (45) () KL = LK K L KL = LK ( ) 6 1 M = 7 12 (46) 10

BM = ( 4 ) 21 31 16, MB = ( 14 ) 13 27 26 (47) BM MB 21 (47) B, M 2 2 22 3 4 * 5 42 m n A m n AA n K KK KK n (KK)K K(KK) 14 KKK 3 3 3 = 3 3 K 2 KK, K 3 KKK (48) m K K m KK K }{{} m (49) 23 B 2, B 3, B 4, K 2, K 3, K 4 43 (determinant) 2 N det N N 11 N 22 N 12 N 21 (50) det N N ( ) N N det B = 3 1 ( 2) 4 = 11 (51) 3 P = P 11 P 12 P 13 P 21 P 22 P 23 (52) P 31 P 32 P 33 *5 11

3 det P P 11 P 22 P 33 P 11 P 23 P 32 + P 12 P 23 P 31 P 12 P 21 P 33 + P 13 P 21 P 32 P 13 P 22 P 31 (53) 4 (50) (53) det(kl) = (det K)(det L) (54) 24 25 2 M det M 2 3 (1) ( 1 ) 3 2 1 (2) ( 1 + i ) 0 0 1 i (3) ( 1 ) i i 1 (4) (5) 2 1 0 1 2 1 0 3 2 2 1 3 6 3 9 1 7 2 26 (54) (1) 2 B, M (54) (2) 2 (54) (3) 3 (54) (4) 3 (54) 44 (Trace) ( ) ( ) n Q trq n trq Q ii = Q 11 + Q 22 + + Q nn (55) i=1 trb = 3 + 1 = 4 (56) 12

* 6 27 28 2 M trm 2 3 (1) ( 1 ) 3 2 1 (2) ( 1 + i ) 0 0 1 i (3) ( 8 ) 2i i 1 (4) (5) 2 1 0 1 2 1 0 3 2 2 1 7 6 3 0 1 7 2 45 n K KI = IK = K (57) 1, I E 1 I Identity E Eigen *7 n n I n, E n I n n 1 0 0 0 0 1 0 0 I n = 0 0 1 0 0 0 0 1 (58) 1 0 { 1 (i = j) δ ij = 0 (i j) (59) i, j 1 n *8 n δ ij I I n I ij δ ij *6 2 2 *7 (8) E *8 1 n 1 n 13

ai a 0 0 0 0 a 0 0 ai = 0 0 a 0 0 0 0 a ai 1 I (60) (ai)a = a(ia) = aa (61) 29 30 2 3 ( ) 1 0 I 2 =, I 0 1 3 = 1 0 0 0 1 0 0 0 1 1 n (1) (2) (3) (4) (5) n n n n i=1 δ ij δ ik δ kj i=1 δ ii j=1 δ ij a j δ ij δ ij δ ij 31 (ai)a (61) 46 n R RR = R R = I n (62) R R () R R 1 R R 1 1 (inverse) f 1 (x) 1 a 1 aa 1 = a 1 a = 1 (63) (62) a 1 = 1 a (64) R 1 1 R 1 1 1 R 11 R 1j 1 R 1 1 1 R i1 R ij 1 1 1 R n1 R nj R 1n R in R nn (65) 14

1 R 2 N N 1 = 1 det N ( ) N22 N 12 N 21 N 11 (66) det N 0 (67) (=) 3 3 4 32 33 NN 1 = N 1 N = I 2 B 1 (1) B 1 (62) (2) B 1 (66) (1) 34 2 *9 (1) (2) (3) ( ) 1 0 0 8 ( ) 1 + i 2 1 1 i ( 2 ) 9 3 6 (4) ( 2 ) 1 1 2 (5) ( 2 ) 1 6 3 (6) ( i ) 6i 1 5 i 35 3 P 5 x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = y 1 a 21 x 1 + a 22 x 2 + + a 2n x n = y 2 a n1 x 1 + a n2 x 2 + + a nn x n = y n (68) *9 15

a 11 x 1 + a 12 x 2 + + a 1n x n y 1 a 21 x 1 + a 22 x 2 + + a 2n x n = y 2 a n1 x 1 + a n2 x 2 + + a nn x n y n (69) a 11 a 12 a 1n x 1 y 1 a 21 a 22 a 2n A =, x = x 2, y = y 2 a n1 a n2 a nn x n y n (70) (69) Ax = y (71) n n A * 10 n 1 (=n )x n 1 (=n )y A x x = A 1 y (72) A 3 2 { ( ) ( ) ( ) 2x 1 + 3x 2 = 6 2 3 x1 6 = 4x 1 5x 2 = 3 4 5 x 2 3 ( ) 2 3 det = 2 ( 5) 3 4 = 22 0 (74) 4 5 (73) ( ) 1 2 3 = 1 4 5 22 ( 5 ) 3 4 2 (75) (73) ( x1 x 2 ) = ( 2 3 4 5 ) 1 ( ) 6 = 1 3 22 ( ) 39 18 (76) { 2x 1 + 3x 2 = 6 4x 1 + 6x 2 = 12 ( ) ( ) 2 3 x1 = 4 6 x 2 ( ) 6 12 (77) det ( ) 2 3 = 2 6 3 4 = 0 (78) 4 6 *10 A ( ) 16

(77) 2 (77) 2x 1 + 3x 2 = 6 (79) 1 (79) (x 1, x 2 ) = (1, 1), (6, 2) (77) { 2x 1 + 3x 2 = 6 4x 1 + 6x 2 = 11 ( ) ( ) ( 2 3 x1 6 = 4 6 x 2 11) (78) 0 (80) 2 (80) 0 = 1? (81) (80) (43) (44) n n n n n n 3 n 10 1000 1 36 2 (1) { 6x 1 + 2x 2 = 13 4x 1 x 2 = 4 (2) { 4x 1 3x 2 = 1 16x 1 + 12x 2 = 4 (3) { x 1 = 5 8x 1 + 6x 2 = 11 (4) { 3x 1 + 3x 2 = 6 x 1 + x 2 = 11 (5) { 1x 1 + 3x 2 = 0 3x 1 + 7x 2 = 0 (6) { 3x 1 + 6x 2 = 20 4x 1 6x 2 = 7 37 3 3x 1 + x 2 = 3 x 1 2x 2 + x 3 = 3 x 2 3x 3 = 6 17

6 2 r θ y y y r θ O x g(θ) x r x 1 2 r ( x r = y) (82) r θ r ( ) ( ) ( ) ( ) x r cos θx sin θy cos θ sin θ x = y = = sin θx + cos θy sin θ cos θ y = g(θ) r (83) g(θ) θ 2 ( ) cos θ sin θ g(θ) sin θ cos θ (84) 2 g(θ) g(θ 1 )g(θ 2 ) = g(θ 2 )g(θ 1 ) = g(θ 1 + θ 2 ) (85) [g(θ)] T g(θ) = g(θ)[g(θ)] T = I 2 (86) n S S T S = SS T = I n (87) S n g(θ) θ 2 n n (87) n n 3) 3 2 18

38 39 (83) g(θ) (85),(86) 40 3 3 (1) 3 z θ g z (θ) cos θ sin θ 0 g z (θ) sin θ cos θ 0 (88) 0 0 1 (2) x ϕ g x (ϕ) y ψ g y (ψ) * 11 (3) g y (ψ)g z (θ)g x (ϕ) (4) g y (ψ)g z (θ) g z (θ)g y (ψ) (5) g y (ψ)g z (θ) g z (θ)g y (ψ) e iθ = cos θ + i sin θ (89) z = x + iy = re iθ, r = x 2 + y 2 (90) z xy Im z = x + iy y r = x 2 + y 2 θ O x Re 2 1 cos θ + i sin θ = e iθ ω(θ) (91) z = x + iy ω(θ)z = (cos θx sin θy) + i(sin θ + cos θy) (92) *11 ϕ ψ 19

(83) z ω(θ) 2 2 g(θ) g(θ) ω(θ) xy 7 1 2 3 20

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