A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $$ are also mentioned.
%0 Journal Article
%1 moler1973algorithm
%A Moler, C. B.
%A Stewart, G. W.
%D 1973
%I Society for Industrial & Applied Mathematics (SIAM)
%J SIAM Journal on Numerical Analysis
%K 65f15-numerical-eigenvalues-eigenvectors
%N 2
%P 241--256
%R 10.1137/0710024
%T An Algorithm for Generalized Matrix Eigenvalue Problems
%U https://doi.org/10.1137%2F0710024
%V 10
%X A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $$ are also mentioned.
@article{moler1973algorithm,
abstract = {
A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned.},
added-at = {2020-07-13T08:53:30.000+0200},
author = {Moler, C. B. and Stewart, G. W.},
biburl = {https://www.bibsonomy.org/bibtex/203496d0de4af8769cc793f2f267d67c8/gdmcbain},
doi = {10.1137/0710024},
interhash = {f5fdabd1370b85b3beedfeeef6eb4f2a},
intrahash = {03496d0de4af8769cc793f2f267d67c8},
journal = {{SIAM} Journal on Numerical Analysis},
keywords = {65f15-numerical-eigenvalues-eigenvectors},
month = apr,
number = 2,
pages = {241--256},
publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
timestamp = {2020-07-13T08:54:03.000+0200},
title = {An Algorithm for Generalized Matrix Eigenvalue Problems},
url = {https://doi.org/10.1137%2F0710024},
volume = 10,
year = 1973
}