condition to be a symmetric matrix

Question

Let us consider two matrices $X$ and $Y$ where both $X$ and $Y$ are $n \times k$ and $X \neq Y$. It is clear that $X'X$ is a symmetric matrix. What I want to know is whether $X'Y$ is always non symmetric when $Y \neq X$ and its proof when it is true. If not, what condition is required for $Y$ and $X$ to have symmetric $X'Y$?

Answer 1

The product $X'Y$ may not be symmetric. As far as I know, without further information about the structure of $X$ and $Y$, there is no substantially different method to verify symmetry beyond performing the product and verifying if the resulting matrix is symmetric.

On the other hand, if $X$ is a scalar multiple of $Y$, then $X'Y$ is symmetric. This is only a sufficient condition. However, it becomes a necessary condition when $X$ and $Y$ are non-zero row vectors.