9.1 Generalized Eigenvalue Problem
The generalized eigenvalue problem for the matrix pair is the problem of finding the pairs and the vectors such that
The scalars are called the generalized eigenvalues and the vectors are the generalized eigenvectors. Mathematically, if is zero or, computationally, if it is close to zero, then the corresponding eigenvalue is set to be . If are the columns of the eigenvector matrix , then Eq. (9.1) can be written in the matrix form as
If the determinant of the matrix does not vanish identically (as a polynomial of ), then the matrix pair is said to be regular; otherwise it is called singular. If the pair is regular and B is nonsingular, then are the eigenvalues and are the eigenvectors of the matrix . If the pair is regular, but B is singular, then there are finite generalized eigenvalues and infinite eigenvalues, where is the degree of the determinant of .
Generalized eigenvalues and eigenvectors.
The functions GeneralizedEigenvalues and GeneralizedEigenvectors compute, respectively, the generalized eigenvalues and eigenvectors; while the function GeneralizedEigensystem computes both generalized eigenvalues and eigenvectors.
If , then the generalized eigenvalue problem is said to be ill-posed (or degenerate). In this case, the functions GeneralizedEigenvectors and GeneralizedEigensystem return unevaluated with a warning message. If the generalized eigenvalue problem is not ill-posed, then there are exactly eigenvalues, including the infinite ones, counting multiplicities.
Make sure the application is loaded.
Here is a pair of 2 × 2 matrices.
Here are the two pairs of scalars associated with the generalized eigenvalues. Both eigenvalues are finite.
Since matrix is nonsingular, the generalized eigenvalues can be computed as the eigenvalues of the matrix .
The generalized eigenvalues are the same as the eigenvalues of .
These are the generalized eigenvectors.
This verifies that the eigenvalues and eigenvectors satisfy Eq. (9.2).