LA_GGEVX computes for a pair of real or complex matrices
the generalized eigenvalues in the form of scalar pairs (
and, optionally, the left and/or right generalized eigenvectors.
A generalized eigenvalue of the pair is, roughly speaking, a scalar of the form such that the matrix is singular. It is usually represented as the pair , as there is a reasonable interpretation of the case (even if ).
A right generalized eigenvector corresponding to a generalized eigenvalue is a vector such that . A left generalized eigenvector is a vector such that , where is the conjugate-transpose of .
The computation is based on the (generalized) real or complex Schur form of . (See LA_GGES for details of this form.)
Optionally, LA_GGEVX also computes a balancing transformation (to improve the conditioning of the eigenvalues and eigenvectors), reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors. The balancing transformation consists of a permutation of rows and columns and/or a scaling of rows and columns.