A generalized eigenvalue of a pair of 
matrices 
is a scalar 
for which there is a nonzero vector 
satisfying
 |
The vector 
is called a generalized eigenvector of the matrix pair. The problem of finding such
pairs 
is called a generalized eigenvalue problem.
If 
is a nonsingular matrix, the generalized eigenvalues
are the ordinary eigenvalues of 
. More generally, when the polynomial 
is not identically zero,
the finite generalized eigenvalues are its roots, which
satisfy
 |
This formulation includes many problems with constraints or nonstandard inner products without explicitly forming the matrix
inverse of 
.
The term generalized eigenvector is also used for a different concept associated with a single matrix.
