Eigenvalues after Involution

Question

Consider the product of two $n \times n$ matrices $UA$, where $U$ is involutory and $A$ has eigenvalues $\lambda_i$ with $i = 1, \dots, n$.

What can I say about the eigenvalues of $UA$?

Answer 1

This is only a partial answer.
I personally doubt whether something definite can be said for the most general case, however for some cases we can observed patterns.

For example assume that $A$ is diagonalizable (for example it has $n$ distinct eigenvalues) and $U$ commutes with $A$. Because involution matrix is always diagonalizable and commutation means that both matrices are simultaneously diagonalized then we can write

$UA=VD_UV^{-1}VD_AV^{-1}=VD_U D_AV^{-1}$

So in this case eigenvalues of $UA$ are products of eigenvalues of $U$ and $A$.

Eigenvalues of $U$ are $\pm1$ so the eigenvalues of $UA$ have the same absolute value as eigenvalues of $A$, but they can have a changed signs.