Given two matrices and the generalized eigenvalue problem is to find nonzero vectors and a number such that The matrix is called a matrix pencil, usually designated by The characteristic...

Given two matrices

and

the generalized eigenvalue problem is to find nonzero vectors

and a number

such that

The matrix

is called a matrix pencil, usually designated by

The characteristic equation for the pencil

is det
and the generalized eigenvalues are the roots of the characteristic polynomial. See

for a discussion of the problem.

How does this problem relate to the standard eigenvalue problem

Show that the degree of the characteristic equation is less than or equal to

Give an example where the degree is less than

In general, when is the degree guaranteed to be less than

Assume that
is nonsingular. Show that

is an eigenvalue of

with associated eigenvector

if and only if

is an eigenvalue of

⁻¹

with associated eigenvector