How can I find a matrix which is orthogonal to another matrix?

I have a matrix A, and want to find the matrix B which is orthogonal to A. How can I do it in Matlab?

e.g.  A=[1,2;2,3;3,4]

Thank you in advance!

[Torsten]

Define "orthogonal" for rectangular matrices.

Best wishes

Torsten.

[Yang Yue]

Thank you! Yes, I mean orthogonal to a rectangular matrix. Is that what you mean?

[Torsten]

?

[Torsten]

In the above case you want to find a row vector v such that

v*[1;2;3] = v*[2;3;4] = 0

?

Is this correct ?

Best wishes

Torsten.

[Yang Yue]

Actually I am not quite sure. I want to find a matrix/vector which is orthogonal to a matrix.

[Torsten]

The only solution I can think of is null(A), the kernel of the matrix.

help null

Best wishes

Torsten.

[Yang Yue]

I will have a look. Thank you very much Torsten!

[Jan]

@Yang Yue: You have repeated some times now, that you want a matrix "orthogonal to a rectangular matrix", but you did not define this expression. Orthogonality of matrices is known for square matrices. So please explain mathematically the wanted properties of the matrix.

[Yang Yue]

Thank you. I read in a paper that matrix B "is the orthogonal matrice to A". I may misunderstood the meaning...

Answer 1

[Yang Yue]

Dear all,

I found the solution. It's called "orthogonal complement". The code here should be

null(alpha','r')

Thank all of your contributions!

Best wishes,

Yang

Answer 2

[Jan]

Maybe you mean the QR decomposition:

A = [1,2; 2,3; 3,4]
[Q, R] = qr(A)

Now Q is an orthogonal matrix, or in other words: Q*Q' == eye(3) . But it is not "orthogonal to A". But please explain, what this exactly means. Perhaps:

A * B == eye(3)

or

B * A == eye(2)

?

[Yang Yue]

Thanks for the answer! I mean is there a matrix B, B*A=0?

[Yang Yue]

Now I am thinking I mean B*A==eye(2)...

[Stephen23]

"I mean is there a matrix B, B*A=0?"

Orthogonal matrices (which are square) are defined by the property Q*Q'=I and Q'*Q=I, so how is zero relevant to the question?

[Yang Yue]

So I think I was wrong... How can I get the matrix B which "is the orthogonal matrice to A"?

[Torsten]

Z = null(A)

gives you the matrix orthogonal to A.

https://de.mathworks.com/help/matlab/ref/null.html

Best wishes

Torsten.

[Yang Yue]

I've tried, but it returns to an empty matrix. I think I need to read more things about "orthogonal".

[Torsten]

For your matrix, only x=[0; 0] satisfies A*x=0.

Check the result with A = [1 2 3; 2 3 4].

Best wishes

Torsten.

[Yang Yue]

Oh I see. Thank you again!

Best wishes, Yang

Answer 3

[Christine Tobler]

The concept of orthogonality for a matrix is defined for just one matrix: A matrix is orthogonal if each of its column vectors is orthogonal to all other column vectors and has norm 1. The concept of two matrices being orthogonal is not defined.

Here's a similar question on math.stackexchange, perhaps one of the answers there would be helpful?

[Yang Yue]

Thank you for the answer. I think I found something. What I want to find is called "orthogonal complement of a subset".