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I am looking for a method of solving eigenvalue problems of the form (I apologize if I gave a misleading title):

\begin{align} \mathbf{A}x + \lambda\mathbf{B}x-c = 0 \end{align}

In particular, I am looking to solve them numerically in matlab, but knowing the proper name for this type of problem would be super helpful!

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I guess you are trying to solve $(A-\lambda B)x=C $ for some non-zero $x$. For this to exist the vector $C $ needs to be in the column space of the matrix on the left. So this then becomes a matter of solving for $\lambda $ that makes the rank of the relevant matrices be the same. Have you tried?

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