# An Algorithm for Generalized Matrix Eigenvalue Problems.

@article{Moler1973AnAF,
title={An Algorithm for Generalized Matrix Eigenvalue Problems.},
author={Cleve B. Moler and G. W. Stewart},
journal={SIAM Journal on Numerical Analysis},
year={1973},
volume={10},
pages={241-256}
}
• Published 1 April 1973
• Computer Science
• SIAM Journal on Numerical Analysis
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.
1,014 Citations
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## References

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• Computer Science
• 1972
The spectrum of $Ax - \lambda Bx = 0$ consists of stable and unstable eigenvalues, which undergo, respectively, small and large changes in response to small changes in A and B. The algorithm isolates
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This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required.
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Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of