Solve the matrix with semi-orthogonal projection $A=V^\top X V$

Question

$V \in \mathbb{R}^{m \times n}$ is a semi-orthogonal matrix, i.e., $V^\top V = I$, with $m > n$. With the following equation: $$A = V^\top X V $$ $A, V$ are given, how to get a closed-form solution of $X$ ?

Discuss:

• $V$ is not a square, so can't take the inverse.
• if extend $V$ to a square form $U$ (orthogonal matrix), and how to construct $U$ (make it invertible)