matrix equation equality

Question

Let $\mathbf{A}$ and $\mathbf{B}$ be real symmetric and square matrices. Let $\mathbf{C}$ be a positive semi-definite matrix and $\mathbf{X}$ any matrix. Let $\mathbf{ACA}=\mathbf{X}$ and $\lambda^2\mathbf{BCB}=\mathbf{X}$. The question is what is the relation between $\mathbf{B}$ and $\mathbf{A}$ and if and when it can be other than $\mathbf{B}=\lambda^{-1}\mathbf{A}$.

Answer 1

The short answer to the second part of your question is: Always, as $B = - \lambda^{-1} A$ is also always a solution. I am assuming that $\lambda$ is real and non-zero.

However, I suspect that is not what you wanted to hear. It is however difficult to say something more concrete without further conditions. For example, for any orthogonal matrix $O$ that commutes with both $C$ and $X$, $B = \pm\lambda^{-1} O A O^T$ are two possible solutions.