Eigenvalues of a symmetric complex matrix

Question

I have a symmetric complex matrix (not Hermitian). Could you tell me about a method to calculate its eigenvalues? I want to choose a method that makes most use of the symmetric property. I know that, for example, the Givens method is useful for symmetric matrices. But it is for symmetric "real" matrices. I searched and found that there are some extensions of it, but the papers were hard for me to understand. I would also be grateful if you introduce some relevant reference. Thanks.

Answer 1

I misread the initial question assuming your were going for Hermitian matrices. It’s not clear to me that this is straightforward.

Taking an example from here $\begin{pmatrix} 2i & 1 \\ 1 & 0 \end{pmatrix}$ isn’t even diagonalizable.

Since you can’t diagonalize it, straightforward Givens is doomed to fail. I suppose there are possible workarounds. That source describes a method using the Lanczos Method.

In general Gibbons has you iteratively cancel out off diagonal terms by applying rotation matrices. For applying this to conplex symmetric matrices, we have to extend this by applying general unitary matrices.

In addition to the simple rotation between two columns, you can also add a complex single column rotation where you multiply a column by $e^{i \theta}$ and the corresponding row by $e^{-i\theta}$. This can help you turn any off diagonal terms into real terms. Then you can apply the standard Gibbs rotation. You can even combine these two matrices into a single unitary rotation between two columns if you have a way of representing that.

A completely separate way to apply Gibbons algorithm to Hermitian matrices is to turn your $n$ by $n$ Hermitian matrix into a $2n$ by $2n$ symmetric matrix. You can replace each value $a+bi$ with $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$

This will give you $2n$ eigenvalues because each of the original $n$ eigenvalues will appear twice.