generate orthogonal matrix function

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,

[David Goodmanson]

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.

Answer 1

[Bruno Luong]

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

[David Goodmanson]

The OP does want square A and B, though ... so you might have to do something like

A = rand(5,4);
A4 = A(:,4);
A = [A*rand(4,4) A4]
B =  rand(5,5);

and proceed from there.

[Bruno Luong]

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

[Bruno Luong]

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