An algorithm to determine if two matrices have common eigenvalues

Given two real lower Hessenberg matrices
A
and
B
of order
n
and
m (m \leq n)
, respectively, a symmetric matrix of order
n
is constructed such that whenever
S
is nonsingular,
A
and
B
do not have an eigenvalue in common. When
S
is singular, its nullity, is the same as the number of common eigenvalues between
A
and
B
. A well-known classical result on the relative primeness of two polynomials and the associated Bezoutian matrix is included as a special case.