# Symmetrizing matrix properties

A symmetrizer $P$ is a $n\times n$ symmetric matrix such that for a $n\times n$ matrix $A$ it holds that $AP=PA^T$. There exists a symmetrizer for any square matrix, and in general it is not unique. Furthermore, if $A$ has complex eigenvalues, then there does not exist a positive definite $P$ that symmetrizes $A$.

I am looking for more information (or literature) on existence and properties of symmetrizers. Particularly:

Are there conditions on $A$ that ensure that a positive (semi-)definite $P$ exists? I am particularly interested in the case $A\in\mathbb{R}^{n\times n}$.

Are there conditions on the existence of symmetrizer, that symmetrizes two distinct matrices $A_1, A_2 \in \mathbb{R}^{n\times n}$?

What can be said about the spectrum of $PA$?

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I am also working on symmetrizers. I have three literature links I would like to point out to you: