Journal of the Operations Research Society of China ›› 2018, Vol. 6 ›› Issue (3): 467-472.doi: https://doi.org/10.1007/s40305-017-0177-z

所属专题： Continuous Optimization

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  • 出版日期:2018-09-30 发布日期:2018-09-30

Spectral Properties and Optimality for Elementary Matrices

Ricardo Biloti1 · João Daniel Palma Ramos2 ·Jin-Yun Yuan3   

  1. 1 Department of Applied Mathematics, University of Campinas & INCT-GP,Campinas 13083-859, Brazil
    2 Instituto Kumon de Educação Ltda., São Paulo 04006-002, Brazil
    3 Department of Mathematics, Federal University of Paraná, Curitiba 81531-980, Brazil
  • Online:2018-09-30 Published:2018-09-30
  • Supported by:

    The work of the first and third authors was partially supported by National Council for Scientific and Technological Development (CNPq), Brazil.

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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.

Key words: Symmetric and triangular (ST) decomposition ·, Non-symmetric system ·Elementary matrix ·, Householder transformation

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