$A$ is known and diagonal, $C$ is known and symmetric and $B$ is symmetric. All matrices are positive semi-definite and of full rank.
2026-04-18 23:24:56.1776554696
Is it possible to uniquely identify matrix $B$ if $C=BAB$ and under the following terms?
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I will assume that $A$ is invertible, since otherwise, the answer is a clear no. Rewrite the equation as $A^{1/2}CA^{1/2}=(A^{1/2}BA^{1/2})^2$, where $A^{1/2}$ is the unique positive definite square root of $A$. But then we must have $A^{1/2}BA^{1/2}=(A^{1/2}CA^{1/2})^{1/2}$, and hence $B=A^{-1/2}(A^{1/2}CA^{1/2})^{1/2}A^{-1/2}$.