31 4 MATLAB A, B R 3 3 A = , B = mat_a, mat_b >> mat_a = [-1, -2, -3; -4, -5, -6; -7, -8, -9] mat_a =
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1 3 4 MATLAB 3 4. A, B R , B = mat_a, mat_b >> mat_a = [-, -2, -3; -4, -5, -6; -7, -8, -9] mat_a = >> mat_b = [-9, -8, -7; -6, -5, -4; -3, -2, -] mat_b = A + B, A B, AB mat_a + mat_b mat_a - mat_b mat_a * mat_b
2 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) O zeros( ) n zeros(n) >> zeros(2, 2) >> zeros(2) 0 0
3 >> zeros(3) ones(, ) n zeros(n) >> ones(3, ) >> ones(3) rand( ), rand( ) >> rand(3,), rand(3, )
4 34 4 MATLAB >> rand(3) A, B R , B = A + 2B 2. A A 2 + 2B 4.2 A a ij A(i, j) >> [, 2, 3; 4, 5, 6; 7, 8, 9] >> A(, 3) 3 >> A(3, )
5 A(, :) A(:, ) (:) >> A(, :) 2 3 >> A(:, 3) ( ) : >> A(:2, 2:3) >> A(3:-:, :)
6 36 4 MATLAB 3 >> A(:, 3:-:) v = 2, v 2 = 4, v 3 = 3 5 >> v = [; 2; 3], v2 = [4; 5; 6], v3 = [7; 8; 9] = P = [v v 2 v 3 ] v = 2 3 v2 = v3 = >> P = [v, v2, v3] P =
7 ( ) 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...
8 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 ([ ]) =, rank MATLAB rank >> rank([, 0; 0, 0]) = 2 >> rank([-5, 0, 5; -5, 0, 5;, 2, 0]) A
9 a P(2, 3)A b AP(2, 3) c R(, 2; )A 2. B B [ ] 3 B = a A = b A 2 = A B [ ] 2, B = 2 [ MATLAB A ( ), B ( ) inv ˆ(-) >> [, 2; 2, ] ] 2 2 >> B = [-2, 6;, -3] B = >> inv(a)
10 40 4 MATLAB 3 >> inv(b) : Inf Inf Inf Inf >> Aˆ(-) B AA = A I >> A * inv(a) 0 0 >> inv(a) * A C D C = , D = /2 /3 /4 /2 /3 /4 /5 /3 /4 /5 /6 /4 /5 /6 /7
11 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 = x = [2 ] T \ ( ) >> [, 2; 2, ]
12 42 4 MATLAB >> b = [4; 5] b = 4 5 >> x = A \ b x = 2 Ax b >> A * x x x 2 x 3 = (determinant) ( ) a a 2 a 2 a 22 = a a 22 a 2 a 2
13 (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 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 = A B >> [, 2; 3, 4] ] >> B = [5, 6; 7, 8] B =
14 44 4 MATLAB 3 >> det(a), det(b) A B = AB >> det(a) * det(b) >> A * B >> det(a * B) AB = A B 4.7 p Frobenius
15 ( 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 [ ] A = max( + 5, ) = max(6, 6) = 6 A = max( + 3, ) = max(4, 8) = 8 λ (AA T ) = , λ 2 (AA T ) = A 2 = max( λ (AA T ), λ 2 (AA T ) ) = A F A F = = 44 MATLAB norm
16 46 4 MATLAB 3 norm(mat_a) norm(mat_a, ) norm(mat_a, inf ) norm(mat_a, fro ) >> [-, -3; -5, -3] >> norm(a), norm(a, ), norm(a, inf ), norm(a, fro ) A [ 2 2 ] A, A, A F
17 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 [ ] Ã = ae 2 (Ã) re2(ã) >> abs(a - short_a).0e-09 * >> abs((a - short_a)./ short_a).0e-09 * >> norm(a - short_a)
18 48 4 MATLAB e-0 >> norm(a - short_a) / norm(a).7930e H. H H 2. H H = /2 /3 /2 /3 /4 /3 /4 /5 H = H re 2 ( H )
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