Name of matrix which is invariant in trace

Question

Let $X$ and $A$ be two real matrices, such that $$\mbox{tr}(XX^T)=\mbox{tr}(XAX^T),$$ where $A\neq I$.

I like to know any information of matrix $A$ and which matrices $A$ can hold the above equation. Is $A$ some particular matrix?

Answer 1

This answer gives only a sufficient condition.

The equation can be rewritten as $\operatorname{tr}(X^TX)=\operatorname{tr}(AX^TX)$

$\operatorname{tr}(X^TX -AX^TX)=0$
$\operatorname{tr}((I-A) X^TX )=0$

$X^TX$ is a symmetric matrix so it is sufficient that $I-A$ be a skew-symmetric matrix, according to Wikipedia.