3 4 MATLAB 3 4. A, B R 3 3 2 3 4 5 6 7 8 9, B = mat_a, mat_b >> mat_a = [-, -2, -3; -4, -5, -6; -7, -8, -9] 9 8 7 6 5 4 3 2 mat_a = - -2-3 -4-5 -6-7 -8-9 >> mat_b = [-9, -8, -7; -6, -5, -4; -3, -2, -] mat_b = -9-8 -7-6 -5-4 -3-2 - A + B, A B, AB mat_a + mat_b mat_a - mat_b mat_a * mat_b
32 4 MATLAB 3 mat_a * vec_v mat_b * vec_w I eye( ) n I n eys(n) >> eye(2, 2) 0 0 >> eye(2) 0 0 >> eye(3, 3) 0 0 0 0 0 0 O zeros( ) n zeros(n) >> zeros(2, 2) 0 0 0 0 >> zeros(2) 0 0
4. 33 0 0 >> zeros(3) 0 0 0 0 0 0 0 0 0 ones(, ) n zeros(n) >> ones(3, ) >> ones(3) rand( ), rand( ) >> rand(3,), rand(3, ) 0.2785 0.5469 0.9575 0.9572
34 4 MATLAB 3 0.4854 0.8003 >> rand(3) 0.49 0.7922 0.0357 0.428 0.9595 0.849 0.957 0.6557 0.9340 4. A, B R 4 4 3 2 9 2 3 2 7 0 4 5 2 2 3 4 0, B = 4 2 7 3 2 8 2 9 0 4 2 2 3 2 0. 3A + 2B 2. A 2 3. 3A 2 + 2B 4.2 A a ij A(i, j) >> [, 2, 3; 4, 5, 6; 7, 8, 9] 2 3 4 5 6 7 8 9 >> A(, 3) 3 >> A(3, )
4.2 35 7 A(, :) A(:, ) (:) >> A(, :) 2 3 >> A(:, 3) 3 6 9 ( ) : >> A(:2, 2:3) 2 3 5 6 2 3 5 6 >> A(3:-:, :) 7 8 9 4 5 6 2 3
36 4 MATLAB 3 >> A(:, 3:-:) 3 2 6 5 4 9 8 7 3 v = 2, v 2 = 4, v 3 = 3 5 >> v = [; 2; 3], v2 = [4; 5; 6], v3 = [7; 8; 9] 6 7 8 = P = [v v 2 v 3 ] v = 2 3 v2 = 4 5 6 v3 = 7 8 9 >> P = [v, v2, v3] P = 4 7 2 5 8
4.3 37 3 6 9 4.3 4. ( ) P(i, j), Q(i; c), R(i, j : c) [] ) P(i, j) i j (i j) : i. j.......... i 0 P(i, j) =......... j 0... Q(i; c) i c (c 0) i..... Q(i; c) =. i c...
38 4 MATLAB 3 R(i, j; c) i j c (c 0) R(i, j; c) = j...... i c....... c 3 P(i, j), Q(i; c), R(i, j; c) 4. ( ) c 0 P(i, j) = P(j, i) (4.) ( Q(i; c) = Q i; ) (4.2) c R(i, j; c) = R(i, j; c) (4.3) (rank) rank ([ 0 0 0 ]) =, rank 5 0 5 5 0 5 2 0 MATLAB rank >> rank([, 0; 0, 0]) = 2 >> rank([-5, 0, 5; -5, 0, 5;, 2, 0]) 2 4.2. A 3 2 2 0 0
4.4 39 a P(2, 3)A b AP(2, 3) c R(, 2; )A 2. B B [ ] 3 B = 9 3 3. 3 4 a A = 4 2 3 b A 2 = 4 5 3 6 7 3 4.4 A B [ ] 2, B = 2 [ 2 6 3 MATLAB A ( ), B ( ) inv ˆ(-) >> [, 2; 2, ] ] 2 2 >> B = [-2, 6;, -3] B = -2 6-3 >> inv(a) -0.3333 0.6667 0.6667-0.3333
40 4 MATLAB 3 >> inv(b) : Inf Inf Inf Inf >> Aˆ(-) -0.3333 0.6667 0.6667-0.3333 B AA = A I >> A * inv(a) 0 0 >> inv(a) * A 4.3 0 0 C D C = 2 3 2 2 3 3 3 3, D = /2 /3 /4 /2 /3 /4 /5 /3 /4 /5 /6 /4 /5 /6 /7
4.5 4 4.5 n A, n b Ax = b n x [ 2 2 ] [ x x 2 ] = [ 4 5 ] A A x = A b >> [, 2; 2, ] 2 2 >> b = [4; 5] b = 4 5 >> x = inv(a) * b x = 2.0000.0000 x = [2 ] T \ ( ) >> [, 2; 2, ]
42 4 MATLAB 3 2 2 >> b = [4; 5] b = 4 5 >> x = A \ b x = 2 Ax b >> A * x 4 5 4.4 3 0 3 0 3 x x 2 x 3 = 8 2 5 4.6 (determinant) 2 4.2 ( ) a a 2 a 2 a 22 = a a 22 a 2 a 2
4.6 43 (i, j) ij (i j n ) n a a 2 a n a 2 a 22 a 2n... a n a n2 a nn n = ( ) i+j ij a ij (j =, 2,..., n) = i= n ( ) i+j ij a ij (i =, 2,..., n) 3 i = a a 2 a 3 3 a 2 a 22 a 23 = ( ) +j j a j a 3 a 32 a 33 j= a = a 22 a 23 a a 2 a 23 a a 32 a 2 + a 2 a 22 33 a 3 a 3 33 a 3 a 32 j= = a a 22 a 33 + a 2 a 23 a 3 + a 3 a 2 a 32 a 3 a 22 a 3 a 2 a 2 a 33 a a 23 a 32 (4.4) 4.2 ( ) n A, B AB = A B MATLAB det [ ] [ 2 5 6, B = 3 4 7 8 A B >> [, 2; 3, 4] ] 2 3 4 >> B = [5, 6; 7, 8] B = 5 6 7 8
44 4 MATLAB 3 >> det(a), det(b) -2-2.0000 A B = AB >> det(a) * det(b) 4.0000 >> A * B 9 22 43 50 >> det(a * B) 4.0000 AB = A B 4.7 p Frobenius
4.7 45 4.3 ( p ) A C n n A p = max x 0 Ax p x p p p A = max x 0 A 2 = max x 0 A = max x 0 Ax x Ax 2 x 2 = Ax x = max j = max i p p n a ij i= max λ i (A T A) ( λ i (A) A i ) i n j= a ij A, B Ax p A p x p AB p A p B p Frobenius F p A C m n A F A F = m n a ij 2 (4.5) i= j= A R 2 2 [ 3 5 3 ] A = max( + 5, 3 + 3 ) = max(6, 6) = 6 A = max( + 3, 5 + 3 ) = max(4, 8) = 8 λ (AA T ) = 22 2 85, λ 2 (AA T ) = 22 + 2 85 A 2 = max( λ (AA T ), λ 2 (AA T ) ) = 22 + 2 85 A F A F = 2 + 3 2 + 5 2 + 3 2 = 44 MATLAB norm
46 4 MATLAB 3 norm(mat_a) norm(mat_a, ) norm(mat_a, inf ) norm(mat_a, fro ) >> [-, -3; -5, -3] - -3-5 -3 >> norm(a), norm(a, ), norm(a, inf ), norm(a, fro ) 6.3592 6 8 6.6332 4.5. A [ 2 2 ] A, A, A F
4.8 47 2. B B = [ i i 2 + i 3i ] B, B, B F 4.8 ae p (Ã) re p (Ã) ae p (Ã) = A (A) p A (A) p re p (Ã) = (A O) A p A (A) p = ae(ã) ( O) (4.6) A R n n [ 2 π sin π/3 3 ] 0 0 Ã [ ].4423562 3.4592654 Ã = 0.8660254038.732050808 ae 2 (Ã) re2(ã) >> abs(a - short_a).0e-09 * 0.373 0.402 0.056 0.43 >> abs((a - short_a)./ short_a).0e-09 * 0.2638 0.306 0.080 0.2489 >> norm(a - short_a)
48 4 MATLAB 3 6.6277e-0 >> norm(a - short_a) / norm(a).7930e-0 4.6 H. H H 2. H H = /2 /3 /2 /3 /4 /3 /4 /5 H = 9 36 30 36 92 80 30 80 80 H re 2 ( H )