# Numerical eigenbasis for a unitary matrix

Do you know what numerical software computes an eigenvector basis for a unitary matrix?

Say I have a unitary matrix $U$. If its eigenvalues are simple (no multiplicities), then for instance Matlab computes an eigenbasis for $U$. However, if there are some eigenvalues with multiplicities, in the subspace for the eigenvalues with multiplicity the software does not find independent eigenvectors. If a matrix is symmetric or Hermitian, Matlab is programmed to output an eigenbasis (even if there are eigenvalues with multiplicities). No such thing for unitary matrices - as fas as I know.

I found a way to avoid this: if $λ$ is an eigenvalue with multiplicity, then I can form the matrix $B=A−λ⋅1$ and find the nullity of $B$. The only problem is that doing this for each possible eigenvalue is slow. I wonder if there is a better solution.

If that makes a difference, I can assume that my unitary matrix is real.

The question was also asked here https://cstheory.stackexchange.com/q/27874/ and here https://stackoverflow.com/q/27533637/ (no answer).

• Crossposting (especially only 18 hours after you posted on SO!) is frowned upon, I think this question is far better suited to SO or math.SE than physics - because nothing in this question is about physics except that unitary matrices are common in physics.
– ACuriousMind
Dec 18, 2014 at 14:47
• This question appears to be off-topic because it is about numerical methods not specific to physics.
– ACuriousMind
Dec 18, 2014 at 14:48
• If the geometric/algebraic multiplicity of the eigenvalues is the same, then there is an eigenspace of dimension > 1 for eigenvalues with multiplicity > 1, and Matlab will return two linearly independent vectors in this eigenspace. Is that enough? Dec 18, 2014 at 16:23
• The original problem is this: I have a state, and I need to decompose it into each eigenspace. Thus, I think that only two independent eigenvectors in a subspace with dimension let's say 6, are not enough. Dec 18, 2014 at 16:34
• Interesting. Do you have an example matrix? Dec 18, 2014 at 19:31

Say I have a unitary matrix $U$. If its eigenvalues are simple (no multiplicities), then for instance Matlab computes an eigenbasis for $U$. However, if there are some eigenvalues with multiplicities, in the subspace for the eigenvalues with multiplicity the software does not find independent eigenvectors.