How to solve the matrix equation $ABA^{-1}=C$ with $\operatorname{Tr}(A)=a$

Question

I have the following matrix equation: $$ABA^{-1}=C$$ with $B$ and $C$ given and $A$ unknown. The constraint on $A$ is $\operatorname{Tr}(A)=a$ with $a\in\mathbb{R}$. The matrices are $N\times N$.

Answer 1

You can rewrite the equation as: $$ AB = CA $$ and using the Kronecker product matrix equation identity $$ \mathrm{vec}(AXB) = (B^T\otimes A)\mathrm{vec}(X) $$ you get $$ (B^T\otimes I_N - I_N\otimes C)\mathrm{vec}(A) = 0, $$ where $I_N$ is an $N\times N$ identity matrix. Together with the trace constraint you have $N^2+1$ equations and $N^2$ unknowns so the existence of the solution is not always guaranteed. For $\alpha = 0$, if the solution exists it would be any element of the null space of the matrix $$ \left[ \begin{array}{c} B^T\otimes I_N - I_N\otimes C \\ \mathrm{vec}(I_N)^T \end{array}\right]. $$