Constrained eigenvalue problem

Question

what is the typical way of solving an eigenvalue problem if you have additional constraints??

Let's say for example

$$ \left(\begin{matrix}a && b && c\\ d && e &&f \\ g && h && i \end{matrix}\right) \left(\begin{matrix}c_1 \\ c_2 \\ c_3\end{matrix}\right)=\lambda\left(\begin{matrix}c_1 \\ c_2 \\ c_3\end{matrix}\right)$$

with a constraint given by e.g. $$c_2=Ac_1.$$ I feel like I should simply add a first column/row multiplied by $A$ to second column/row ??

All are complex numbers.

Answer 1

I'm guessing that the $A$ is a constant. For your example above, you should just be able to substitute $c_2=Ac_1$. In which case, you should be able to solve the equation below by your normal methods.

$$ \left(\begin{matrix}a && b && c\\ d && e &&f \\ g && h && i \end{matrix}\right) \left(\begin{matrix}c_1 \\ Ac_1 \\ c_3\end{matrix}\right)=\lambda\left(\begin{matrix}c_1 \\ Ac_1 \\ c_3\end{matrix}\right)$$