The matrix $A$ has dimension $m \times m$ and $B$ has dimension $n \times n$ with $m > n$. The matrix $X$ is hence non-square, but matrices $A$ and $B$ are square.
Is there some elegant way to derive the matrix $X$ if the matrices $A$ and $B$ are known ?
Thanks!
With $A$ $m \times m$ and $B$ $n \times n$, the entries of $X^T A X - B$ correspond to $n^2$ quadratic polynomials in $mn$ unknowns. "Generically" we might expect this system to have Hilbert dimension $mn - n^2$, i.e. there should be a family of (complex) solutions with $mn-n^2$ free parameters.
In practice, I might use Gröbner basis techniques to find solutions.
Caution: these solutions can be horribly complicated, even if $m$ and $n$ are fairly small.