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I need to construct real-valued matrices with specific complex eigenvalues. I have seen the companion matrix, which sort of does my job, but there are some other desirable properties as well, so I'm looking for a more general construction. If anyone could shed some light, that'd be awesome.

The eigenvalues will be distinct. They will lie on the unit circle. Preferably, I could choose the eigenvectors as well (I know some restrictions will apply, but it's fine). It would be amazing, if the matrix had as few zeros as possible.

So, given I am asking for a lot, I will appreciate the most general construction.

I have seen this answer (), but I wonder if there is a method that provides greater freedom over the matrix.

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1 Answer

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Build $M$ with $2 \times 2$ block matrices along the diagonal as in the linked question. Then for any invertible matrix $A$ the matrix $AMA^{-1}$ will have the eigenvalues you want. You can design $A$ to get the other properties you care about.

The original $M$ is likely to be the one with the most zeroes.

A random square matrix $A$ will be invertible (the probability that the determinant is $0$ is $0$) and I'm pretty sure $AMA^{-1}$ will have no $0$ entries with probability $1$.

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