Generalized Nonsymmetric Eigenproblems (GNEP)

Given a matrix pair , where and are square
matrices, the **generalized nonsymmetric eigenvalue problem** is to find
the **eigenvalues** and corresponding
**eigenvectors**
such that

Note that these problems are equivalent with and if neither nor is zero. In order to deal with the case that or is zero, or nearly so, the LAPACK routines return two values, and , for each eigenvalue, such that and .

More precisely, and are called

are called

Sometimes the following, equivalent, notation is used to refer to the generalized eigenproblem for the pair : The object , where is a complex scalar variable, is called a

If the determinant of is identically zero for all values of , the eigenvalue problem is called

The generalized nonsymmetric eigenvalue problem can be solved via the

where and are orthogonal matrices, is upper triangular, and is an upper quasi-triangular matrix with and diagonal blocks, the blocks corresponding to complex conjugate pairs of eigenvalues of . In the

where and are unitary and and are both upper triangular.

The columns of and are called

More formally, let and be a conformal partitioning with respect to the cluster of eigenvalues in the (1,1)-block of , i.e. where and both have columns, and and below are both ,

Then subspaces and form a pair of (left and right) deflating subspaces associated with the cluster of , satisfying and [39,40]. It is possible to order the generalized Schur form so that has any desired subset of eigenvalues, taken from the set of eigenvalues of .

As for the standard nonsymmetric eigenproblem, two pairs of drivers are provided, one pair focusing on the generalized Schur decomposition, and the other pair on the eigenvalues and eigenvectors as shown in Table 2.6:

- LA_GGES:
a simple driver that computes all or part of the
generalized Schur decomposition
of , with optional
ordering of the eigenvalues;

- LA_GGESX:
an expert driver that can additionally compute condition
numbers for the average of a selected subset of eigenvalues,
and for the corresponding pair of deflating subspaces of and ;

- LA_GGEV:
a simple driver that computes all the generalized
eigenvalues of , and optionally the left or right
eigenvectors (or both);

- LA_GGEVX: an expert driver that can additionally balance the matrix pair to improve the conditioning of the eigenvalues and eigenvectors, and compute condition numbers for the eigenvalues and/or left and right eigenvectors (or both).