Matrix pencil

Summary

A matrix pencil is a family of matrices $A - \lambda B$ that depends linearly on a parameter $\lambda$. Its eigenvalues are the values of $\lambda$ for which $A - \lambda B$ is singular, which leads to the generalised eigenvalue problem $Av = \lambda Bv$. Pencils arise in vibration problems, in differential-algebraic equations and in control theory, and they are classified by the Kronecker canonical form. More general pencils allow higher powers of $\lambda$; the linear ones are the most common.