= Generalized eigenvalue
{disambiguate=matrix pencil}
{title2=$Av=\lambda Bv,\quad v\ne0$}
= Generalized eigenvalue
{synonym}
A <generalized eigenvalue (matrix pencil)> 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 $A=\operatorname{diag}(2,6)$ and $B=\operatorname{diag}(1,2)$, 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>.<Eigenvalues and Schur Forms in the LAPACK Users' Guide>[Generalized eigenvalue definition and symmetric-definite versus singular cases]{ref}
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