I am interested in properties of the matrix product $AB$ where the matrix $A\in\mathbb{R}^{n\times n}$ is positive definite (i.e., $(Av,v)>0$ for all non-zero vectors $v\in\mathbb{R}^n$) and where matrix $B\in\mathbb{R}^{n\times n}$ is symmetric (i.e., $(Bu,v)=(Bv,u)$ for all $u,v\in\mathbb{R}^n$).
I know that we can not expect positiveness or symmetry without additional assumptions but are we at least able to exlude skew-symmetry of the product (except for the trivial case $B=0$ of course)? I have tried to investigate this and expect it to be the case but I wasn't able to show it (or to find a counter-example).
Hence my question:
Is it possible for the product $AB$ of a positive definite matrix $A$ and a symmetric matrix $B\neq0$ to be skew-symmetric?
Note: I am specifically thinking about $n=3$ but have formulated the question for general dimension $n\in\mathbb{N}$.
Best Answer
When $B\ne0$, no, because $AB$ is similar to the real symmetric matrix $A^{1/2}BA^{1/2}$.