Compute an Orthogonal Matrix
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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
5 个评论
Torsten
2019-4-11
Your matrix A can't be diagonalized using an orthogonal matrix U.
Jan
2019-4-11
Your A is a singular matrix.
David Goodmanson
2019-4-15
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
2019-4-15
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
2019-4-15
编辑：David Goodmanson
2019-4-15
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
回答（1 个）
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.
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