On Factorization and Calculation of Determinant of Block Matrices with Triangular Submatrices

Abstract

In this paper, we consider some block matrices of dimension $nm\times{nm}$ whose components are triangular matrices of dimension $n\times{n}$. We prove that the determinant of such block matrices is determined only by the diagonal elements of their submatrices and that this determinant is expressed as the multiplication of some subdeterminants. If the components of dimension $n\times{n}$ are all diagonal matrices, then we prove that such a block matrix can be written as a product of simpler matrices. Besides, we investigate the eigenvalues, the adjoint, and the inverse of such block matrices.

Keywords

• Block matrix
• determinant
• triangular matrix
• trigonometric system
• Wronskian
• factorization

References

1. M. Saadetoglu, Ş. M. Dinsev, Inverses and determinants of $n\times n$ block matrices, Mathematics 11 (17) (2023) 3784 12 pages.
2. M. El-Mikkawy, A note on a three-term recurrence for a tridiagonal matrix, Applied Mathematics and Computation 139 (2--3) (2003) 503-511.
3. N. D. Buono, G. Pio, Nonnegative matrix tri-factorization for co-clustering: An analysis of the block matrix, Information Sciences 301 (2015) 13-26.
4. Y. Zhang, M. Zhang, Y. Liu, S. Ma, S. Feng, Localized matrix factorization for recommendation based on matrix block diagonal forms, in: D. Schwabe, V. Almeida, H. Glaser (Eds.), Proceedings of the 22nd International Conference on World Wide Web, Rio de Janeiro, 2013, pp. 1511-1520.
5. E. Kamgnia, L. B. Nguenang, Some efficient methods for computing the determinant of large sparse matrices, Revue Africaine de Recherchéen Informatique et Mathématiques Appliquées 17 (2014) 73-92.
6. R. Schachtner, G. Pöppel, E. W. Lang, Towards unique solutions of nonnegative matrix factorization problems by a determinant criterion, Digital Signal Processing 21 (4) (2011) 528-534.
7. J. T. Jia, J. Wang, T. F. Yuan, K. K. Zhang, B. M. Zhong, An incomplete block-diagonalization approach for evaluating the determinants of bordered k-tridiagonal matrices, Journal of Mathematical Chemistry 60 (8) (2022) 1658-1673.
8. M. S. Solary, From matrix polynomial to the determinant of block Toeplitz-Hessenberg matrix, Numerical Algorithms 94 (3) (2023) 1421-1434.

9. P. Sakkaplangkul, N. Chuenjarern, Computational efficiency for calculating determinants of block matrices, RMUTP Research Journal Sciences and Technology 18 (1) (2024) 38-46.
10. J. Liu, J. Bi, M. Li, Secure outsourcing of large matrix determinant computation, Frontiers of Computer Science 14 (6) (2020) Article Number 146807 12 pages.
11. D. Grinberg, P. J. Olver, The $n$ body matrix and its determinant, SIAM Journal on Applied Algebra and Geometry 3 (1) (2019) 67-86.
12. J. Y. Shao, H. Y. Shan, L. Zhang, On some properties of the determinants of tensors, Linear Algebra and Its Applications 439 (10) (2013) 3057-3069.
13. K. Neymeyr, M. Sawall, On the set of solutions of the nonnegative matrix factorization problem, SIAM Journal on Matrix Analysis and Applications 39 (2) (2018) 1049-1069.
14. M. Cé, L. Giusti, S. Schaefer, Local factorization of the fermion determinant in lattice QCD, Physical Review D 95 (3) (2017) 034503 13 pages.
15. W. E. Boyce, R.C. DiPrima, Elementary differential equations, 10th Edition, Wiley, New York, 2012.
16. S. Lipschutz, Linear algebra, 4th Edition, McGraw-Hill, New York, 2009.
17. U. Kaya, Wronski determinant of trigonometric system, Journal of Advanced Mathematics and Mathematics Education 5 (1) (2022) 1-8.
18. R. Bellman, Introduction to matrix analysis, Society for Industrial and Applied Mathematics, Philadelphia, 1987.