Symmetric matrix as sum

Question

Let $B$ be a symmetric matrix, and $A$ an invertible matrix (or course both square matrices with the same order). Show that there is a matrix $X$ such that $$B= AX^T + A^T X$$

Answer 1

JUST A COMMENT: Are you sure you didn't miss any additional assumptions?

I think this is not possible in general. For instance, take $A=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix}1 & 1 \\ 1 & 0 \end{pmatrix}$. Writing $X=\begin{pmatrix}x_1 & x_2 \\ x_3 & x_4 \end{pmatrix}$, your equation is equivalent to the linear system \begin{align*} x_2-x_3&=1\\ x_2-x_3&=0, \end{align*} which obviously has no solution.

In general, if $A$ is symmetric, but $A^{-1}B$ is not (see the example above), then your equation cannot have a solution: If $A$ is symmetric, then $B=AX^T+A^TX=A(X^T+X)$, and this implies $X^T+X=A^{-1}B$. The left hand side is symmetric, but the right hand side is not.