# If $S$ is symmetric positive definite and $SA$ symmetric, is then $A$ symmetric?

We are given real matrices $$S$$ and $$A$$. We know that $$S$$ is symmetric positive definite and that $$SA$$ is symmetric. Is A necessarily symmetric then?

I've figured out that if $$A$$ is symmetric, then $$S$$ and $$A$$ must commute. I've tried finding a $$2 \times 2$$ and a $$3 \times 3$$ counterexample, as it seemed to me that this is not generally true, but I couldn't find any.

• Is $S$ also assumed symmetric? Commented Jun 5, 2021 at 18:34
• @BartMichels Yes. I will add that to my question. Commented Jun 5, 2021 at 18:40
• Start by finding a symmetric positive definite $S$ and symmetric $B$ that don't commute. Then let $A=S^{-1}B$.
– user932138
Commented Jun 5, 2021 at 18:53
• Yes, following O.Peters suggestion, you can ''easily'' find a counterexample. He's $B$ is yours $SA$ Commented Jun 5, 2021 at 19:04

Let $$S = \begin{bmatrix}2 & 0 \\ 0 & 1 \end{bmatrix}$$, $$A = \begin{bmatrix}1 & 1/2 \\ 1 & 2 \end{bmatrix}$$. Then $$SA = \begin{bmatrix}2 & 1 \\ 1 & 2 \end{bmatrix}$$ is symmetric, but $$A$$ is not.