Find Eigenvalues of multiplied Matrices when the corresponding Eigenvalues are known

Question

I am trying to find the eigenvalues or in particular the largest eigenvalue of a transformation which consists of two matrices: $A = B C$.

Assuming I know the EV of both matrices $B$ and $C$, is there a general relation to the EV of $A$?

Thank you very much in advance. Roman

Answer 1

Not really. Consider $$ B=\begin{bmatrix}1&0\\0&2\end{bmatrix},\ \ C=\begin{bmatrix}1&0\\0&2\end{bmatrix}. $$ Then the eigenvalues of $A=BC$ are $1$ and $4$. But you change $C$ to $$ C=\begin{bmatrix}2&0\\0&1\end{bmatrix}, $$ then the eigenvalues of $BC$ are $2,2$.

Even worse, if you take any two matrices $B',C'$ and form the block matrices $$ B=\begin{bmatrix}B'&0\\0&0\end{bmatrix}, \ \ C=\begin{bmatrix}0&0\\0&C'\end{bmatrix}, $$ then $BC=0$ no matter what the eigenvalues of $B$ and $C$ are.