# Inhomogeneous eigenvalue problem

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!

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?