Spectral Properties and Optimality for Elementary Matrices
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Abstract
A generalization of the Householder transformation, renamed as elementary matrix by A.S. Householder: Unitary transformation of a nonsymmetric matrix, J. ACM, 5(4), 339–342, 1958, was introduced by LaBudde (Math Comput 17(84):433–437, 1963) as a tool to obtain a tridiagonal matrix similar to a given square matrix. Some of the free parameters of the transformation can be chosen to attain better numerical properties. In this work, we study the spectral properties of the transformation. We also propose a special choice for free coefficients of that transformation to minimize its condition number. The transformation with such suitable choice of parameters is called optimal.
Keywords
Symmetric and triangular (ST) decomposition Non-symmetric system Elementary matrix Householder transformationMathematics Subject Classification
Primary 65F10 Secondary 65F15Notes
Acknowledgements
The authors thank Prof. F. Bazán for his helpful comments and references [7, 8]. The first author thanks the support of the sponsors of the Wave Inversion Technology Consortium.
The authors are in debit with Gene Golub for his valuable contributions to this paper, specially for Theorem 2.2.
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