Determinant, inertia, and inverse of a class of irreducible tridiagonal matrices
-
Cristian Conde
,
Ezequiel Dratman
Abstract
In this paper, we study a class of irreducible tridiagonal matrices and analyze their fundamental properties. We derive explicit formulas for their determinant, providing insights into their spectral behavior. Additionally, we establish results concerning their inertia and obtain closed-form expressions for their inverse. These findings contribute to the broader understanding of structured matrices and have potential applications in various fields of mathematics and other sciences.
Funding statement: The present work was supported by PIP2022-2024 GI-11220210100392CO, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.
Acknowledgements
The authors sincerely thank the referee for the careful reading of the manuscript and for the valuable comments and suggestions, which have helped us to improve the quality and clarity of the paper.
References
[1] R. B. Bapat, An interlacing theorem for tridiagonal matrices, Linear Algebra Appl. 150 (1991), 331–340. 10.1016/0024-3795(91)90178-YSearch in Google Scholar
[2] L. Brugnano and D. Trigiante, Tridiagonal matrices: Invertibility and conditioning, Linear Algebra Appl. 166 (1992), 131–150. 10.1016/0024-3795(92)90273-DSearch in Google Scholar
[3] K. Castillo, C. M. da Fonseca and J. Petronilho, On Chebyshev polynomials and the inertia of certain tridiagonal matrices, Appl. Math. Comput. 467 (2024), Article ID 128497. 10.1016/j.amc.2023.128497Search in Google Scholar
[4] C. M. da Fonseca and V. Kowalenko, Eigenpairs of a family of tridiagonal matrices: Three decades later, Acta Math. Hungar. 160 (2020), no. 2, 376–389. 10.1007/s10474-019-00970-1Search in Google Scholar
[5] M. E. A. El-Mikkawy, On the inverse of a general tridiagonal matrix, Appl. Math. Comput. 150 (2004), no. 3, 669–679. 10.1016/S0096-3003(03)00298-4Search in Google Scholar
[6] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins Stud. Math. Sci., Johns Hopkins University, Baltimore, 2013. Search in Google Scholar
[7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University, Cambridge, 1985. 10.1017/CBO9780511810817Search in Google Scholar
[8] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University, Cambridge, 1991. 10.1017/CBO9780511840371Search in Google Scholar
[9] S. Hovda, Closed-form expression for the inverse of a class of tridiagonal matrices, Numer. Algebra Control Optim. 6 (2016), no. 4, 437–445. 10.3934/naco.2016019Search in Google Scholar
[10] U. Kamps, Chebyshev polynomials and least squares estimation based on one-dependent random variables, Linear Algebra Appl. 112 (1989), 217–230. 10.1016/0024-3795(89)90597-1Search in Google Scholar
[11] D. Kulkarni, D. Schmidt and S.-K. Tsui, Eigenvalues of tridiagonal pseudo-Toeplitz matrices, Linear Algebra Appl. 297 (1999), no. 1–3, 63–80. 10.1016/S0024-3795(99)00114-7Search in Google Scholar
[12] G. Labahn and T. Shalom, Inversion of Toeplitz matrices with only two standard equations, Linear Algebra Appl. 175 (1992), 143–158. 10.1016/0024-3795(92)90306-USearch in Google Scholar
[13] J. W. Lewis, Inversion of tridiagonal matrices, Numer. Math. 38 (1981/82), no. 3, 333–345. 10.1007/BF01396436Search in Google Scholar
[14] G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl. 13 (1992), no. 3, 707–728. 10.1137/0613045Search in Google Scholar
[15] P. Schlegel, The explicit inverse of a tridiagonal matrix, Math. Comp. 24 (1970), Paper No. 665. 10.1090/S0025-5718-1970-0273798-2Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
