As the question above, if $A,B$ are square (complex) diagonal matrices and $C$ is square real symmetric and $$ A=BC, $$ Does this imply that $C$ must also be diagonal?
Some context: This arose in a physics problem where I started with complex square matricies $A’$ and $B’$ simultaneously diagonalisible in the same basis and $C’$ real diagonal, s.t. $A’=B’C’$. So my approach was that in the eigenbasis of $A’$ and $B’$ we get the above form and I got stuck. All I want to find is an expression for the eigenvalues of $A$ in terms of the eigenvalues of $B$ and $C$ if that is possible.
Edit: $A$ and $B$ are invertible and thus non-zero.