Institution: Stanford University, Department of Computer Science

Title: Kronecker's canonical form and the QZ algorithm

Author: Wilkinson, James Hardy

Date: April 1979

Abstract: In the QZ algorithm the eigenvalues of Ax = $\lambda$Bx are computed via a reduction to the form $\tilde{A}$x = $\lambda \tilde{B}$x where $\tilde{A}$ and $\tilde{B}$ are upper triangular. The eigenvalues are given by ${\lambda}_i$ = $a_{ii}$/$b_{ii}$. It is shown that when the pencil $\tilde{A}$ - $\lambda \tilde{B}$ is singular or nearly singular a value of ${\lambda}_i$ may have no significance even when $\tilde{a}_{ii}$ and $\tilde{b}_{ii}$ are of full size.

http://i.stanford.edu/pub/cstr/reports/cs/tr/79/735/CS-TR-79-735.pdf