Given A and B as square matrices is A solvable in ABA = C?

Question

Say we have 3 square matrices A, B, C and equation ABA = C. Can A be solved? If A were solvable for square matrices, would A be solvable for non-square matrices?

Answer 1

For general square matrix, the answer is no. This is because there are usually infinitely many square roots for a given matrix.

However, if $B$ is positive definite and $C$ is positive semi-definite, then there is an unique positive semi-definite $A$ which satisfies $ABA = C$.

Recall for any positive semi-definite matrix $M$, there is an unique positive semi-definite matrix $N$ which satisfies $N^2 = M$. We will use $\sqrt{M}$ to refer to this unique square root. In terms of this square root operation over positive semi-definite matrices, we have

$$\begin{align} ABA = C &\implies (\sqrt{B}A\sqrt{B})^2 = \sqrt{B}A\sqrt{B}^2A\sqrt{B} = \sqrt{B}C\sqrt{B}\\ & \implies \sqrt{B}A\sqrt{B} = \sqrt{\sqrt{B}C\sqrt{B}}\\ & \implies A = \sqrt{B}^{-1} \sqrt{\sqrt{B}C\sqrt{B}}\sqrt{B}^{-1} \end{align}$$