A General Matrix Eigenvalue Algorithm

Abstract

A general $QR$-type process called the $VZ$ algorithm is presented for the solution of the general matrix eigenvalue problem $ACx = \lambda BDx$. The matrices involved may be rectangular. For appropriate choices of A , B, C and D, we have some of the more familiar types of eigenproblems, and this is reflected in the fact that the $QR$, $QZ$ and $SVD$ algorithms are all special cases of the $VZ$ algorithm. The main emphasis is upon the algorithm’s generality as well as its bearing upon the generalized singular value problem $A^T Ax = \mu ^2 B^T Bx$.