Toeplitz-plus-Hankel matrices with perturbed corners

Document Type : Research Paper

Author

Assistant Professor of Applied Mathematics, Payame Noor University, Po Box 19395-3697 Tehran, IRAN

Abstract

This paper examines suitable borderings and modification techniques for finding some special properties of a class of real heptadiagonal symmetric Toeplitz matrices and anti-heptadiagonal persymmetric Hankel matrices with perturbed corners as the zeros of explicit rational functions. An orthogonal diagonalization, inverse and determinant, and a formula to compute its integer powers for these matrices are shown. Then, these results are expanded for the corresponding Toeplitz-plus-Hankel matrices with perturbed corners.

Keywords

[1] D. Bini and M. Capovani, Sepctral and computational properties of band symmetric Toeplitz matrices, Linear
Algebra Appl. 52/53 (1983), 99–126.
[2] D. Bini and M. Capovaui, Tensor rank and border rank of band Toeplitz matrices, SIAM J. Comput. 16 (1987),
no. 2, 252–258.
[3] J.R. Bunch, C.P. Nielsen and D.C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math.
31 (1978), 31–48.
[4] D. Fasino, Spectral and structural properties of some pentadiagonal symmetric matrices, Calcolo 25 (1988), no.
4, 301–310.
[5] M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), no.
4, 27–31.
[6] G.H. Golub, Some modified matrix eigenvalue problems, SIAM Rev. 15 (1973), no. 2, 318–334.
[7] D.A. Harville, Matrix algebra from a statistician’s perspective, Springer-Verlag, New York, 1997.
[8] C. R. Johnson and R. A. Horn, Topics in matrix analysis, Cambridge University Press, 1991.
[9] R.A. Horn and C.R. Johnson, Matrix analysis, second ed. Cambridge University Press, 2013.
[10] J. Lita da Silva, On anti-pentadiagonal persymmetric Hankel matrices with perturbed corners, Comput. Math.
Appl. 72 (2016), 415–426.
[11] M. Shams Solary, Finding eigenvalues for heptadiagonal symmetric Toeplitz matrices, J. Math. Anal. Appl. 402
(2013), 719–730.
[12] M. Shams Solary, Computational properties of pentadiagonal and anti-pentadiagonal block band matrices with
perturbed corners, Soft Comput. 24 (2020), 301–309.
[13] J. H. Wilkinson, The algebraic eigenvalue problem, Oxford U. P. London, 1965.
Volume 13, Issue 2
July 2022
Pages 3057-3072
  • Receive Date: 07 June 2020
  • Revise Date: 06 April 2022
  • Accept Date: 25 April 2022