generate orthogonal matrix function

Question

mingcheng nie 2023-9-8

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https://ww2.mathworks.cn/matlabcentral/answers/2018381-generate-orthogonal-matrix-function

评论： Bruno Luong 2023-9-8

Hi there,

Assume I have two square matrix A and B with the same size, I want to find a matrix that orthogonal to A but non-orthogonal to B, i.e., AC=0 and BC≠0, where C is the matrix I want.

Is there any function can generate it? I know null(A) can generate the orthogonal matrix of A, but I want to ensure it is non-orthogonal to B meanwhile.

Thanks,

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David Goodmanson 2023-9-8

编辑：David Goodmanson 2023-9-8

Hi MN,

for this to work there will have to be restrictions on A and B. For example, A cannot be of full rank, because if it is, then it has and inverse, in which case AC = 0 --> C=0. So suppose A is not of full rank. Then B must have at least one row that is linearly independent of the rows of A, otherwise AC = 0 --> BC = 0.

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Answer 1

Bruno Luong 2023-9-8

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https://ww2.mathworks.cn/matlabcentral/answers/2018381-generate-orthogonal-matrix-function#answer_1305146

编辑：Bruno Luong 2023-9-8

在 MATLAB Online 中打开

I claim this returns the solution C that is orthogonal to A but not to B (meaning not to all of columns of B)

A = rand(5,4)*rand(4,5);
B =  rand(5,4)*rand(4,5);
AO = null(A.');
ABO = null([A B].');
v = null(ABO'*AO);
if isempty(v)
    fprintf('solution does not exist\n')
    return
end
v1 = v(:,1);
[Q,~,~] = qr([2*v1, eye(size(AO,2))]);
C = AO*Q
C = 5×1
    0.3321
    0.1400
   -0.5466
   -0.1237
    0.7457
% Check
norm( A'*C, 'fro')
ans = 2.3876e-16
norm( B'*C, 'fro')
ans = 1.8113

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Bruno Luong 2023-9-8

Thanks I fix it just a small detail, my algo supposes to work with non-square and square matrices

Bruno Luong 2023-9-8

在 MATLAB Online 中打开

To get the orthogonal of rows A, B, change the first two statements

A = rand(5,4)*rand(4,5);
B =  rand(5,4)*rand(4,5);
AO = null(A);       % change
ABO = null([A; B]); % change
v = null(ABO'*AO);
if isempty(v)
    fprintf('solution does not exist\n')
    return
end
v1 = v(:,1);
[Q,~,~] = qr([2*v1, eye(size(AO,2))]);
C = AO*Q
C = 5×1
    0.4633
   -0.2224
   -0.6358
   -0.4591
    0.3476
% Check
norm( A*C, 'fro')
ans = 4.2999e-16
norm( B*C, 'fro')
ans = 1.5602