I have an invertible matrix $A$ of size $n \times n$ and a matrix $U$ of size $n \times m$, for $m < n$. Matrix $U$ is orthonormal, meaning, the rows are orthonormal vectors. I also have an $m \times m$ diagonal matrix $\Sigma$ with positive values on the diagonal.
I want to find the solution for the equation
$X A U = \Sigma.$
Since $U$ is orthonormal, then $U^T U = I$, and therefore, for $X = \Sigma U^T A^{-1} $, we have:
$X A U = \Sigma U^T A^{-1} A U = \Sigma.$
I would like to know if this is the only solution, and if so, how to show it.