Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access January 18, 2019

Bidiagonalization of (k, k + 1)-tridiagonal matrices

  • S. Takahira EMAIL logo , T. Sogabe and T.S. Usuda
From the journal Special Matrices


In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].


[1] M. El-Mikkawy and F. Atlan, A novel algorithm for inverting a general k-tridiagonalmatrix, Appl.Math. Lett., 32, (2014), 41-47. DOI:10.1016/j.aml.2014.02.01510.1016/j.aml.2014.02.015Search in Google Scholar

[2] C. M. da Fonseca, T. Sogabe, and F. Yilmaz, Lower k-Hessenberg matrices and k-Fibonacci, Fibonacci-p and Pell (p, i) numbers, Gen. Math. Notes, 31, 1, (2015), 10-17.Search in Google Scholar

[3] A. Fukuda, E. Ishiwata, M. Iwasaki, and Y. Nakamura, The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues, Inverse Probl., 25, 1, (2009), 015007. DOI:10.1088/0266-5611/25/1/01500710.1088/0266-5611/25/1/015007Search in Google Scholar

[4] M. H. Gutknecht, Variants of BICGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput., 14, 5, (1993), 1020-1033. DOI:10.1137/091406210.1137/0914062Search in Google Scholar

[5] T. McMillen, On the eigenvalues of double band matrices, Linear Algebra Appl., 431, 10, (2009), 1890-1897. DOI:10.1016/j.laa.2009.06.02610.1016/j.laa.2009.06.026Search in Google Scholar

[6] A. Ohashi, T. Sogabe, and T. S. Usuda, Fast block diagonalization of (k, k0)-pentadiagonal matrices, Int. J. Pure Appl. Math., 106, 2, (2016), 513-523. DOI:10.12732/ijpam.v106i2.1410.12732/ijpam.v106i2.14Search in Google Scholar

[7] T. Sogabe and M. El-Mikkawy, Fast block diagonalization of k-tridiagonal matrices, Appl. Math. Comput., 218, 6, (2011), 2740-2743. DOI:10.1016/j.amc.2011.08.01410.1016/j.amc.2011.08.014Search in Google Scholar

Received: 2018-04-02
Accepted: 2018-09-28
Published Online: 2019-01-18

© by S. Takahira, et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 28.9.2023 from
Scroll to top button