Compute an Orthogonal Matrix

Hi All,

I need your help. Is there any solution in Matlab to compute an orthogonal matrix if the first coulomn of the orthogonal matrix is known.

For example, I want to find an orthonal matrix for matrix A,

A = [1 0 0 0 -1 0;-1 1 0 0 0 0;0 -1 1 0 0 0;0 0 -1 1 0 0;0 0 0 -1 1 0;0 0 0 0 -1 1];

U*A*inv(U) = B

U is an orthogonal matrix with the first coulomn of U being [1;1;1;1;1;1] .

B is a diagonal matrix with all eigenvalues of A on the diagonal.

Thank you very much for your help

[Torsten]

Your matrix A can't be diagonalized using an orthogonal matrix U.

[Jan]

Your A is a singular matrix.

[David Goodmanson]

Hi namo,

should the first row of A be [0 0 0 0 0 -1] instead of [0 0 0 0 -1 0] ? That puts A into a nice looking form and allows a solution like you are talking about.

[Matt J]

U is an orthogonal matrix with the first coulomn of U being [1;1;1;1;1;1] .

The norm of the columns (and the rows) of an orthogonal matrix must be one. So, a column of 1's is impossible. Maybe you mean that the column should be [1;1;1;1;1;1] /sqrt(6).

[David Goodmanson]

Hi Matt / namo

yes that's true, thanks for pointing it out.

In the specific case of the modified A, there is a U of the right form, but I had not noticed before that it is still not quite right because

U'*A*U = B
whereas the question wanted 
U*A*U' = B 

Answer 1

[Matt J]

No, this is generally not possible. When all the eigenvalues of A are distinct, for example, the (orthonormalized) eigenvectors are unique up to sign. That means you cannot arbitrarily specify one column of U.