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I am working on a problem where the following equation came up

$${\bf X}_1{\bf A}{\bf X}_2{\bf A}^T{\bf X}_3{\bf A}-{\bf X}_4={\bf A}$$

where ${\bf A}$ is an arbitrary $n\times n$ and ${\bf X}_i$s are unknown diagonal real matrices. My question is if it is feasible and if there is a computationally tractable way to solve it.

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It might be more feasible to put some constraints on, e.g. set B= (ZAZ) for Z zome diagonal matrix, and try to find solutions to X1BBTB - X2 = B. Gerhard "Ask Me About System Design" Paseman, 2011.03.14 – Gerhard Paseman Mar 14 '11 at 21:52
up vote 5 down vote accepted

Looking at it entry-by-entry, you have $n^2$ equations in $4n$ variables. However, there is some symmetry here since multiplying X1, X2, X3 respectively by scalars whose product is 1 leaves the result unchanged. So there are really just $4n-2$ degrees of freedom rather than 4n. If $n \ge 4$ we have $n^2 > 4n - 2$ so I would expect there to be no solutions in general. If $n \le 3$ we have $n^2 < 4n - 2$ and I would expect there to be infinitely many solutions. Groebner basis methods may be able to find actual solutions if n is small.

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