19 4 30
I
1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23
342 24 343 27 344 29 35 31 351 31 352 33 353 35
1 11 m, n a ij (1 i m, 1 j n) a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n A = a 31 a 32 a m1 a m2 a mn m n (m, n) a 11,a 12,a 13,,a 1n a 11 a 21 a 31 a ij A a 11 a 12 a ij a m1 (1, 1) (1, 2) (i, j) A =(a ij ) i=1,,m; j=1,,n A =(a ij ) 11 (1) (4, 2) (2) ( 4 0 1 2 A = 3 7 6 5 ) 1
(2, 3) (3) a ij = 2j i + j A =(a ij ) i=1,2,3; j=1,2 12 1 A, B A =(a ij ) i=1,,m; j=1,,n, B =(b ij ) i=1,,m; j=1,,n A + B A + B := (a ij + b ij ) i=1,,m; j=1,,n 11 2 1 3 4 0 1 + 1 1 0 1 2 0 = 3 0 3 5 2 1 c ca := (ca ij ) 12 ( 1 2 3 2 4 5 6 ) 11 A B ( ) A B = A +( B) 1 m, n 2
11 ( ) A, B, C (m, n) c, d (1) A + B = B + A (2) (A + B)+C = A +(B + C), c(da) =(cd)a (3) c(a + B) =ca + cb, (c + d)a = ca + da 12 11 ( ) 0 O 0( ) 13 131 (2, 2) ( )( ) ( ) a b α β aα + bγ aβ + bδ := c d γ δ cα + dγ cβ + dδ (m, n) A (n, l) B AB A B 13 ( 2 2 1 5 3 1 ) 1 2 2 4 3 2 3
13 (1) (m, n) A (n, l) B AB (m, l) (2) AB BA 14 AB BA AB, BA 15 132 100 100 1+2+ + 99 + 100 = n = j n=1 j=1 6 1+3+5+ +11= (2m 1) = m=1 5 (2l +1) l=0 16 9 2+6+10+14+ +38= (4n +2) n=0 n n a 1 + a 2 + + a n = a k = a j k=1 k=1 k=1 j=1 n n c( a k )= ca k m m m a k + b j = (a k + b k ) k=1 j=1 k=1 m a 11 + a 12 + + a 1m = a 1j j=1 4
a 11 + a 12 + + a 1n +a 21 + a 22 + + a 2n + = = m ( i=1 +a m1 + a m2 + + a mn n a ij ) (i j ) j=1 n m ( a ij ) j=1 i=1 (j i ) 133 A =(a ij ) (m, n) B =(b ij ) (n, l) a 11 a 12 a 1n b 11 b 12 b 1l a 21 a 22 a 2n b 21 b 22 b 2l AB = a m1 a m2 a mn b n1 b n2 b nl AB (1, 1) c 11 c 11 = a 11 b 11 + a 12 b 21 + + a 1n b n1 = n a 1k b k1 k=1 AB (2, 1) c 21 c 21 = a 21 b 11 + a 22 b 21 + + a 2n b n1 = n a 2k b k1 k=1 AB (1, 2) c 12 c 12 = a 11 b 12 + a 12 b 22 + + a 1n b n2 = n a 1k b k2 k=1 AB (i, j) c ij c ij = n a ik b kj k=1 5
2 ( n ) AB = a ik b kj k=1 i=1,,m; j=1,,l 134 12 A, B (m, n) C (n, l) D (r, m) c R (1) (A + B)C = AC + BC ( ) (2) D(A + B) =DA + DB ( ) (3) c (AB) = (ca)b = A (cb) ( ) (4) (AB)C = A(BC) ( ) (5) AB BA (1) A =(a ij ), B =(b ij ), C =(c ij ) A + B =(a ij + b ij ) ( n (A + B)C = = = k=1 (a ik + b ik )c kj ) ( n ) (a ik c kj + b ik c kj ) k=1 ( n a ik c kj + k=1 = AC + BC ) n b ik c kj ( ) k=1 (2) (1) (3) c R n c a ik b kj = k=1 n (ca ik )b kj = k=1 n a ik (cb kj ) k=1 2 AB (m, l) 6
(4) A =(a ij ) (m, n) B =(b ij ) (n, l) C =(c ij ) (l, p) AB (i, j) (AB) ij (AB) ij = n a ik b kj k=1 (AB)C (i, j) ((AB)C) ij = = = = = l (AB) ir c rj r=1 l n ( a ik b kr )c rj r=1 l k=1 r=1 k=1 n k=1 r=1 n a ik b kr c rj l a ik b kr c rj n a ik ( k=1 r=1 ( ) ( ) l b kr c rj ) ( ) l r=1 b krc rj BC (k, j) A(BC) 14 (n, n) n A, B n AB, BA n AB BA 17 2 AB = BA AB BA n 1 0 E n E n 1( ) 11 n B E n B = BE n = B 7
E =(e ij ) e ij = δ ij (E n B) ij = n e ik b kj = e ii b ij = b ij k=1 BE n 1 1 A n AX = XA = E n n X A X A A 1 18 X 19 (A 1 ) 1 = A 13 A, B n AB (AB) 1 = B 1 A 1 X = B 1 A 1 (AB)X =(AB)(B 1 A 1 )=A(BB 1 )A 1 = E n X(AB) =E n 11 A j 110 (A 1 A 2 A k ) 1 = A 1 k A 1 k 1 A 1 2 A 1 1 8
2 21 211 (1) 0 c eg ( ) ( 1 2 3 2 2 1 2 3 4 5 6 8 10 12 ) (2) c eg ( ) ( 1 2 3 1 + 2 ( 2) 7 8 9 4 5 6 8 10 12 ) (3) eg ( 1 2 3 4 5 6 ) ( 1 2 4 5 6 1 2 3 ) (1)-(3) 21 n E n 21 E n A E n 9
21 1 2 2 3 4 2 2 1 1 1 2 2 2 + 1 ( 3) 0 2 4 2 1 1 1 2 2 3 + 1 ( 2) 0 2 4 (1, 1) 0 3 5 1 2 2 2 1/2 0 1 2 0 3 5 1 0 2 1 + 2 2 and 3 + 2 ( 3) 0 1 2 0 0 1 1 0 2 3 ( 1) 0 1 2 0 0 1 1 0 0 1 + 3 ( 2) and 2 + 3 ( 2) 0 1 0 0 0 1 E 3 22 E 3 2 1 1 3 1 2 1 1 1 212 10
22 ( )( ) ( ) 0 1 1 2 3 4 = = 1 2 1 0 3 4 1 2 ( )( ) ( ) 3 0 1 2 3 6 = = 1 3 0 1 3 4 3 4 ( 0 1 ) ( 3 0 ) 1 0, 0 1 E n E 3 (1) 1 c (c 0) 1 0 0 c 0 0 0 1 0 0 1 0 =: M(1; c) 0 0 1 0 0 1 23 1 0 0 0 3 0 0 0 1 1 2 3 4 5 6 7 8 9 = (2) 2 + 1 c 1 0 0 0 1 0 0 0 1 1 0 0 c 1 0 0 0 1 =: A(2, 1; c) 24 1 0 0 0 1 4 0 0 1 1 2 3 4 5 6 7 8 9 = (3) 1 2 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 =: P (1, 2) 11
25 1 0 0 0 0 1 0 1 0 1 2 3 4 5 6 7 8 9 = 22 M(i; c) E n M(1; 1/c)M(1; c)e n = M(1; c)m(1; 1/c)E n = E n c 0 A(i, j; c) A(2, 1; c) A(2, 1; c)e n = A(2, 1; c) A(2, 1; c)e n = E n P (i, j) P (2, 1)P (1, 2)E n = E n A E n 23 n A A E n B k (k =1, 2,m) B m B m 1 B 2 B 1 A = E n B k P := B m B 1 PA = E n AP = P 1 P (AP )=P 1 (PA)P = E n A 26 21 12
213 24 ( ) n A 1 (1) E n = 1 (2) ( Er O O ), E r r 21 (1) E r r A rank ( ) r = rank A (2) E n rank A = n (3) rank (4) E n A E n 27 21 (4) 3 23 1 2 1 0 0 1 0 0 3 28 2 2 3 1 1 1 1 1 3 23 E n A A 1 E n 25 24 (2) A 3 13
(2) ( Er ) A = O O AX = E n n A ( Er ) A O O 24 n P j,q j ( Er ) P k P 1 AQ 1 Q l = O O ) ) ( 1 2 3 4 )( 0 1 1 0 = ( 2 1 4 3 B := P k P 1 AQ 1 Q l 214 24 25 A A E n ( ) PA = E n (P ) P E n E n X PE n = P X = P 14
26 n A (n, 2n) (A E n ) A E n (A E n ) (E n P ) P A 1 A E n A 29 1 3 5 (1) 2 5 7 2 4 3 (2) 3 2 3 2 2 3 5 2 4 22 221 24 27 ( ) (m, n) A 1 (1) 1 (2) ( Er O O ) (3) ( E m ) 15
(4) ( En O ) x +2y +3z =1 4x +5y +6z = 1 7x +8y +9z = 3 (21) a 11 x 1 + + a 1n x n = b 1 a m1 x 1 + + a mn x n = b m n (22) (21) 1 2 3 A = 4 5 6 7 8 9 1 2 3 1 A =(Ab) = 4 5 6 1 7 8 9 3 (22) 210 1 0 1 2 3 A = 4 1 0 1 0 1 3 1 2 1 222 16
24 x + y =1 2x 3y =1 2x +2y =2 5y =1 y =1/5 x =4/5 28 ( ) A B B A 211 (21) 22 (A b) A = (A b ) A (A b ) ( b) 223 212 x +2z =2 4x y +5z =1 2x + y +3z =4 27 17
1 0 3 1 (A b) 0 1 1 1 0 0 0 3 x +3z =1 y + z =1 0=3 29 ranka = rank(a b) b = 0 0 (i) (1) 1 0 0 1 (A b) 0 1 0 1 0 0 1 3 x =1 y = 1 z =3 (ii) (2) 1 0 1 4 (A b) 0 1 2 1 0 0 0 0 z = t x = t +4 y = 2t +1 18
x y z = t +4 2t +1 t = t 1 2 1 + 4 1 0 t (iii) (2) (A b) 1 0 1 3 1 0 1 4 1 1 0 0 0 0 0 z = t, w = s x = t +3s +1 y =4t + s 1 x y z w = t +3s +1 4t + s 1 t s = t 1 4 1 0 + s 3 1 0 1 + 1 1 0 0 213 (1) 1 0 4 1 0 1 3 1 0 0 0 0 (2) ( 1 0 3 1 4 0 1 1 0 2 ) (3) 1 0 3 0 1 5 0 0 1 (4) 1 0 0 4 1 3 1 0 1 0 1 2 1 1 0 0 1 6 0 5 0 0 0 0 0 0 0 0 19
214 x + y z = 2 (1) 2x +3y 4z = 9 x +2y 3z = 7 x +2y +3z =1 (2) 2x 2y +3z = 1 x 2y +5z =2 3x 3y z =1 (3) 2x + y + bz = 1 2x +5y 3z = a (a, b) 224 (A b) Ax = b b = 0 b 0 ranka = rank(a 0) x = 0 0 x 1 210 (A b) x x = x 0 + x 1, x 0 215 1 2 3 1 4 5 6 2 7 8 9 5 20
3 31 n n =2 ( ) a b A = c d A = ad bc A 0 A ( ) A 1 = 1 d b A c a n 3 n n =2 n 3 32 1 n (1, 2,,n) n (1, 2, 3) (1, 3, 2) (2, 3, 1) (1, 2, 3) (1, 2, 3) (2, 3, 1) (1, 3, 2) (1, 2, 3) n σ (1, 2,,n) sign σ =1 21
sign σ = 1 k (1, 2,,n) sign σ =( 1) k 31 sign (2, 3, 1) = 1 31 31 (1) (2, 1) (2) (3, 2, 1) (3) (2, 4, 1, 3) (4) (5, 1, 2, 3, 4) n S n 32 S 3 = {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)} 32 S 2,S 4 S n n! (n ) 33 n A =(a ij ) i,j=1,,n A ( det A) A = sign (p 1,,p n )a p1 1a p2 2 a pnn (p 1,,p n) S n 22
33 n =2 A = (p 1,p 2 ) S 2 sign (p 1,p 2 )a p1 1a p2 2 = sign (1, 2)a 11 a 22 + sign (2, 1)a 21 a 12 = a 11 a 22 a 21 a 12 33 n =3 4 A n! n! n 34 341 31 B O a nn = a nn B n =3 a 11 a 12 0 a 21 a 22 0 a 31 a 32 a 33 = a 33 a 11 a 12 a 21 a 22 (n =3) A = sign (p 1,p 2,p 3 )a p1 1a p2 2a p3 3 (p 1,p 2,p 3 ) S 3 a 13 = a 23 =0 p 3 p 3 =3 a 33 4 23
a 11 a 12 0 a 21 a 22 0 = sign (p 1,p 2, 3)a p1 1a p2 2a 33 a 31 a 32 a 33 (p 1,p 2,3) S 3 = a 33 sign (p 1,p 2, 3)a p1 1a p2 2 (p 1,p 2,3) S 3 sign (p 1,p 2, 3) = sign (p 1,p 2 ) (p 1,p 2, 3) (p 1,p 2 ) a 11 a 12 0 a 21 a 22 0 = a 33 sign (p 1,p 2 )a p1 1a p2 2 a 31 a 32 a 33 (p 1,p 2 ) S 2 32 34 1 2 0 4 5 0 7 8 9 =9 1 2 4 5 = 9(5 8) = 27 342 (m, n) A =(a ij ) B =(b ij ) A B (n, m) b ij = a ji t A 35 t 1 2 3 4 5 6 7 8 9 = 1 4 7 2 5 8 3 6 9 ( ) 1 4 t 1 2 3 = 2 5 4 5 6 3 6 24
32 (1) t ( t A)=A (2) t (A + B) = t A + t B (3) t (ca) =c t A, c R (4) t (AB) = t B t A (5) A t A ( t A) 1 = t (A 1 ) (1) (3) (4) A =(a ij ) (m, n) B =(b ij ) (n, l) AB (i, j) (AB) ij n (AB) ij = a ik b kj k=1 ( t (AB)) ij = n a jk b ki k=1 ( t A) ij = a ji, ( t B) ij = b ji ( t B t A) ij = n ( t B) ik ( t A) kj = n b ki a jk k=1 k=1 34 (5) 33 ( ) A =(a ij ) t A = A 25
t A (i, j) ( t A) ij ( t A) ij = a ji t A = sign(p 1,p 2,p 3 )( t A) p1 1( t A) p2 2( t A) p3 3 (p 1,p 2,p 3 ) S 3 = sign(p 1,p 2,p 3 )a 1p1 a 2p2 a 3p3 (p 1,p 2,p 3 ) S 3 a 1p1 a 2p2 a 3p3 a q1 1a q2 2a q3 3 a 1p1 a 2p2 a 3p3 = a q1 1a q2 2a q3 3 (p 1,p 2,p 3 ) (1, 2, 3) k (q 1,q 2,q 3 ) k (1, 2, 3) sign(p 1,p 2,p 3 ) = sign(q 1,q 2,q 3 ) (p 1,p 2,p 3 ) (q 1,q 2,q 3 ) (p 1,p 2,p 3 ) S 3 (q 1,q 2,q 3 ) S 3 sign(p 1,p 2,p 3 )a 1p1 a 2p2 a 3p3 = sign(q 1,q 2,q 3 )a q1 1a q2 2a q3 3 = A (p 1,p 2,p 3 ) S 3 (q 1,q 2,q 3 ) S 3 31 B O a nn = a nn B 35 36 4 1 2 3 2 3 0 0 3 =( 3) 4 1 3 2 = 15 33 ( ) 33 26
343 n A =(a ij ) A =(a 1, a 2,,a n ) 1 a 1 = 4 7 1 2 3 A = 4 5 6 7 8 9, a 2 = 2 5 8, a 3 = 3 6 9 a 1, a 2,, a n = sign (p 1,,p n )a p1 1a p2 2 a pnn (p 1,,p n) S n 31 (1) a 1,,λa j + µb,,a n = λ a 1,,a j,,a n + µ a 1,,b,,a n (2) a 1,,a j,,a k,,a n = a 1,,a k,,a j,,a n (1) n =3 a 1, a 2 + b, a 3 = sign (p 1,p 2,p 3 ) a p1 1(a p2 2 + b p2 )a p3 3 (p 1,p 2,p 3 ) S 3 + α R a 1,αa 2, a 3 (2) a 1, a 3, a 2 = (p 1,p 2,p 3 ) S 3 sign (p 1,p 2,p 3 ) a p1 1a p2 3a p3 2 27
a p1 1a p2 3a p3 2 a p1 1a p2 3a p3 2 = a p1 1a p3 2a p2 3 (p 1,p 2,p 3 ) (p 1,p 3,p 2 ) sign (p 1,p 2,p 3 )= sign (p 1,p 3,p 2 ) (p 1,p 2,p 3 ) S 3 (p 1,p 3,p 2 ) S 3 34 (1) 31 (2) 0 a 1,,a j,,a j,,a n =0 (2) 31 (1) a 1,,a j + λa k,,a n = a 1,,a j,,a n (3) 33 1 2 0 1 3 1 5 6 4 1 3 0 1 1 2 6 0 0 0 1 3 13 5 6 = (1, 4) 4 1 3 0 7 13 2 6 7 13 2 6 3 13 5 6 = 4 1 3 0 0 0 0 1 7 13 2 = 3 13 5 31 4 1 3 28
45 13 37 = 49 13 44 (3, 2) 0 1 0 45 37 13 = 49 44 13 0 0 1 = ( 45 44 + 37 49) = 167 37 A =(a 1, a 2, a 3 ) A =3 (1) a 3, 5a 2, a 1 (2) a 2, a 1 + a 3, a 2 a 1 (3) 2a 2 + a 3, 3a 3 + a 1, 4a 1 + a 2 (1), (2) (3) index choice (2, 3, 1) (3, 1, 2) 344 34 n A, B AB = A B 35 (1) (2) A A 0 A AX = E n ( ) X AX = E n =1 AX = A X A 0 29
n =3 AB = a 11 a 12 a 13 a 21 a 22 a 23 b 11 b 12 b 13 b 21 b 22 b 23 a 31 a 32 a 33 b 31 b 32 b 33 = A(b 1, b 2, b 3 ) = (Ab 1,Ab 2,Ab 3 ) AB b 11 0 0 Ab 1 = A 0 + A b 21 A 0 0 0 b 31 1 0 0 = b 11 A 0 + b 21A 1 + b 31A 0 0 0 1 = b 11 a 1 + b 21 a 2 + b 31 a 3, A =(a 1, a 2, a 3 ) Ab j = b 1j a 1 + b 2j a 2 + b 3j a 3 AB = b 11 a 1 + b 21 a 2 + b 31 a 3,b 12 a 1 + b 22 a 2 + b 32 a 3,b 13 a 1 + b 23 a 2 + b 33 a 3 AB = b p1 1a p1,b p2 2a p2,b p3 3a p3 (p 1,p 2,p 3 ) S 3 AB = b p1 1b p2 2b p3 3 a p1, a p2, a p3 (p 1,p 2,p 3 ) S 3 a p1, a p2, a p3 a 1, a 2, a 3 k a p1, a p2, a p3 =( 1) k a 1, a 2, a 3 (p 1,p 2,p 3 ) (p 1,p 2,p 3 ) (1, 2, 3) k 30
sgn(p 1,p 2,p 3 )=( 1) k AB = b p1 1b p2 2b p3 3 sgn(p 1,p 2,p 3 ) a 1, a 2, a 3 (p 1,p 2,p 3 ) S 3 = a 1, a 2, a 3 sgn(p 1,p 2,p 3 )b p1 1b p2 2b p3 3 (p 1,p 2,p 3 ) S 3 36 A =5, B =3 (1) A 2 (2) A 1 (3) (AB) 1 (4) B 1 AB 35 351 n =3 n 3 A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 3 A (1, 1) a 11 ã 11 ã 11 =( 1) 1+1 a 22 a 23 a 32 a 33 A (1, 1) a 11 31
a 23 ã 23 5 ã 23 =( 1) 2+3 a 11 a 12 a 31 a 32 37 3 A 36 n 38 (1) 1 0 1 2 1 3 1 1 3 (2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 35 ( ) 3 A =(a ij ) A = a 11 ã 11 + a 21 ã 21 + a 31 ã 31 A = a 11 ã 11 + a 12 ã 12 + a 13 ã 13 ( ) ( ) 38 ( ) 1 2 1 2 0 5 4 0 3 2 5 = 2( 1) 1+2 4 3 = 2( 6 20) = 52 ( ) 5 a 23 32
( ) A n A = a 11 1 a 12 a 1n 0 0 a n2 a nn + a 21 0 a 12 a 1n 1 0 a n2 a nn + + a n1 0 a 12 a 1n 0 1 a n2 a nn ã 11,,ã n1 1 (n, n) ã ij 0 B 11 B 12 0 1 0 B 21 B 22 0 =( 1) n j ( 1) n i B 11 B 12 B 21 B 22 =( 1) 2n (i+j) B 11 B 12 B 21 B 22 =( 1) i+j B 11 B 12 B 21 B 22 =ã ij 39 352 32 resp resp 0=a 12 ã 11 + a 22 ã 21 + a 32 ã 31 ( ) 33
0=a 11 ã 31 + a 12 ã 32 + a 13 ã 33 ( ) a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 ã 11 + a 21 ã 21 + a 31 ã 31 a 11,a 21,a 31 a 12 a 12 a 13 a 22 a 22 a 23 = a 12 ã 11 + a 22 ã 21 + a 32 ã 31 a 32 a 32 a 33 37 A 3 A =(a ij ) ã 11 ã 12 ã 13 t ã 21 ã 22 ã 23 ã 31 ã 32 ã 33 = ã 11 ã 21 ã 31 ã 12 ã 22 ã 32 ã 13 ã 23 ã 33 = Ã A Ã A 32 ã 11 ã 21 ã 31 a 11 a 12 a 13 A 0 0 ÃA = ã 12 ã 22 ã 32 a 21 a 22 a 23 = 0 A 0 ã 13 ã 23 ã 33 a 31 a 32 a 33 0 0 A AÃ ÃA = AÃ = A E 3 A 0 1/ A ( 1 1 = A( A Ã)A A Ã) =E 3 34
36 ( ) n A A A 0 A 0 A A 1 A 1 = 1 A Ã, Ã A 38 n =2 A =(a ij ) ( A 1 1 a22 a 12 = a 11 a 22 a 12 a 21 a 21 a 11 ) 310 353 a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 32 a 33 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3, x = x 1 x 2 x 3, b = b 1 b 2 b 3 Ax = b 37 A n A 0 A 35
Ax = b n =3 b 1 a 12 a 13 b 2 a 22 a 23 b 3 a 32 a 33 x 1 = b A a 11 b 1 a 13 a 21 b 2 a 23 a 31 b 3 a 33 x 2 = b A a 11 a 12 b 1 a 21 a 22 b 2 a 31 a 32 b 3 x 3 = b A A (x =)A 1 Ax = A 1 b Ãb A 1 = 1 Ã A A Ã, Ãb = x = 1 A Ãb ã 11 ã 21 ã 31 ã 12 ã 22 ã 32 ã 13 ã 23 ã 33 b 1 ã 11 + b 2 ã 21 + b 3 ã 31 b A b 1 a 12 a 13 b 1 ã 11 + b 2 ã 21 + b 3 ã 31 = b 2 a 22 a 23 b 3 a 32 a 33 x 2,x 3 36 b 1 b 2 b 3
39 4x +8y = 13 3x +11y =4 13 8 4 11 x = 4 8 3 11 4 13 3 4 y = 4 8 3 11 37
371-0816 E-mail ken@maebashi-itacjp URL http://umekensakuranejp/kenwiki/indexphp