DOI QR코드

DOI QR Code

ON REDUCTION OF K-ALMOST NORMAL AND K-ALMOST CONJUGATE NORMAL MATRICES TO A BLOCK TRIDIAGONAL FORM

  • ASIL, K. NIAZI (DEPARTMENT OF MATHEMATICS, LORESTAN UNIVERSITY) ;
  • KAMALVAND, M. GHASEMI (DEPARTMENT OF MATHEMATICS, LORESTAN UNIVERSITY)
  • Received : 2019.08.21
  • Accepted : 2019.09.11
  • Published : 2019.09.25

Abstract

This paper examines how one can build a block tridiagonal structure for k-almost normal matrices and also for k-almost conjugate normal matrices. We shall see that these representations are created by unitary similarity and unitary congruance transformations, respectively. It shall be proven that the orders of diagonal blocks are 1, k + 2, 2k + 3, ${\ldots}$, in both cases. Then these block tridiagonal structures shall be reviewed for the cases where the mentioned matrices satisfy in a second-degree polynomial. Finally, for these processes, algorithms are presented.

References

  1. M.GHASEMI KAMALVAND, KH. D. IKRAMOV, A method of congruent type for linear systems with conjugate-normal coefficient matrices. Computational mathematic and physics, Vol. 49, No. 2, 203-216, 2009. https://doi.org/10.1134/S0965542509020018
  2. R. BEVILACQUA, G. M. DEL CORSO, A condensed representation of almost normal matrices. Linear algebra and its applications, 438, 4408-4425, 2013. https://doi.org/10.1016/j.laa.2013.02.004
  3. R. BEVILACQUA, G. M. DEL CORSO, L. GEMIGNANI, Block tridiagonal reduction of perturbed normal and rank structured matrices. Linear algebra and its applications,1-13, 2013.
  4. L. ELSNER, KH. D. IKRAMOV, On a condensed form for normal matrices under finite sequences of elementary unitary similarities. Linear algebra and its applications, 254, 79-98, 1997. https://doi.org/10.1016/S0024-3795(96)00526-5
  5. A. BUNSE-GERSTNER, R. STOVER, On a conjugate-type mathod for solving complex symmetric linear systems. Linear algebra Appl.287, 105-123, 1999. https://doi.org/10.1016/S0024-3795(98)10091-5
  6. M.GHASEMI KAMALVAND, KH. D. IKRAMOV, Low-rank perturbations of normal and conjugate-normal matrices and their condensed forms under unitary similarities and congruences. Computational mathematic and physics, Vol. 33, No. 3, 109-116, 2009.
  7. M. DANA, A. G. ZYKOV, KH.D. IKRAMOV, A minimal residual method for a special class of the linear systems with normal coefficient matrices. - Comput. Math. Math. Phys., 45, 1854-1863, 2005.
  8. R.A. HORN AND C.R. JOHNSON, Matrix Analysis. Cambridge University Press, Cambridge, 1985.
  9. W. KAHAN, Spectra of nearly Hermitian matrices. Proc. Amer. Math. Soc. 48, 11-17, 1975. https://doi.org/10.1090/S0002-9939-1975-0369394-5
  10. J. SUN, On the variation of the spectrum of a normal matrix. Linear algebra Appl, 246, 215-223, 1996. https://doi.org/10.1016/0024-3795(94)00354-8
  11. I. IPSEN, Departure from normality and eigenvalue perturbation bounds. Technical Report TR03-28, NC State University, 2003.
  12. R. VANDEBRIL, G.M. DEL CORSO, An implicit multishift qr-algorithm for Hermitian plus low rank matrices. SIAM J. Sci. Comput. 16, 2190-2212, 2010.
  13. L.G.Y. EIDELMAN, I.C. GOHBERG, Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbation. Numer. orithms 47, 253-273, 2008.
  14. M. PUTINAR, Linear analysis of quadrature domains. III. J. Math. Anal. Appl, 239(1):101-117, 1999. https://doi.org/10.1006/jmaa.1999.6556