Generalized eigenvalue (matrix pencil, $Av=\lambda Bv,v\ne 0$)

A generalized eigenvalue is a scalar $\lambda$ for which $Av=\lambda Bv$ has a nonzero solution $v$. For a regular square pencil this is equivalent, at finite $\lambda$, to $\det (A-\lambda B)=0$.

For $A=\mathrm{diag}(2,6)$ and $B=\mathrm{diag}(1,2)$, the generalized eigenvalues are two and three. If $B$ is invertible, these are the eigenvalues of $B^{-1}A$, although explicitly forming that inverse is not necessary to solve the problem. Singular $B$ can introduce an eigenvalue at infinity, while a singular pencil is a separate case. For symmetric $A$ and positive-definite symmetric $B$, transforming with $B^{-1/2}$ gives a symmetric ordinary eigenproblem and hence real eigenvalues.^[ref]