Purpose

Next: Arguments Up: Generalized Nonsymmetric Eigenvalue Problems Previous: LA_GGES Contents Index

Purpose

LA_GGES computes for a pair of $n\times n$ real or complex matrices the (generalized) real or complex Schur form, the generalized eigenvalues in the form of scalar pairs $(\alpha, \beta)$ , and, optionally, the left and/or right Schur vectors.
If and are real then the real-Schur form is computed, otherwise the complex-Schur form is computed. The real-Schur form is a pair of real matrices such that 1) has block upper triangular form, with $1 \times 1$ and $2 \times 2$ blocks along the main diagonal, 2) has upper triangular form with nonnegative elements on the main diagonal, and 3) and , where and are orthogonal matrices. The $2 \times 2$ blocks of are ``standardized'' by making the corresponding elements of have the form

$\begin{displaymath}\left[ \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right] \end{displaymath}$

The complex-Schur form is a pair of matrices

such that 1)

has upper triangular form, 2)

has upper triangular form with nonnegative elements on the main diagonal, and 3)

and

, where

and

are unitary matrices.
In both cases the columns of

and

are called, respectively, the left and right (generalized) Schur vectors.
A generalized eigenvalue of the pair

is, roughly speaking, a scalar of the form $\lambda = \alpha/\beta$ such that the matrix $A - \lambda B$ is singular. It is usually represented as the pair $(\alpha, \beta)$ , as there is a reasonable interpretation of the case $\beta = 0$ (even if $\alpha = 0$ ).

Next: Arguments Up: Generalized Nonsymmetric Eigenvalue Problems Previous: LA_GGES Contents Index

Susan Blackford 2001-08-19