On two classes of two-parameter Hessenberg matrices

Abdullah Alazemi; Emrah Kılıç; Abdullah Alazemi; Emrah Kılıç

doi:10.3934/math.20261267

AIMS Mathematics

2026, Volume 11, Issue 9: 32223-32244. doi: 10.3934/math.20261267

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Research article

On two classes of two-parameter Hessenberg matrices

Abdullah Alazemi ^{1
,
,},
Emrah Kılıç ²

1.
Department of Mathematics, College of Science, Kuwait University, Sabah Al Salem University City, P. O. Box 5969, Safat 13060, Shadadiya, Kuwait
2.
TOBB University of Economics and Technology, Mathematics Department, Ankara 06560, Turkey

Received: 18 June 2026 Revised: 17 September 2026 Accepted: 24 September 2026 Published: 30 September 2026
MSC : 05A19, 11B37, 15A15, 15A36, 15A57

We investigate two classes of Hessenberg matrices defined by two parameters. For both classes, we explicitly compute the determinant, the inverse matrix, and the LU-factorization. For the matrix of the first class, we explicitly determine its spectrum, whereas the derivation of the eigenvalues for the second class is left as an open problem. Furthermore, we derive recurrence relations, various identities, and combinatorial sums satisfied by the determinants of matrices of both classes. Another contribution of this work is the formulation of the inverse matrix, the determinant, and the LU-factorization using the determinants of all leading principal Hessenberg submatrices. By specializing the parameters, we establish relationships between these Hessenberg determinants and certain integer or polynomial sequences as well as a factorization of the generalized Fibonacci numbers related to the first class.
- Hessenberg matrices,
- eigenvalue,
- recurrence,
- factorization,
- Zeilberger algorithm
Citation: Abdullah Alazemi, Emrah Kılıç. On two classes of two-parameter Hessenberg matrices[J]. AIMS Mathematics, 2026, 11(9): 32223-32244. doi: 10.3934/math.20261267

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Abstract

We investigate two classes of Hessenberg matrices defined by two parameters. For both classes, we explicitly compute the determinant, the inverse matrix, and the LU-factorization. For the matrix of the first class, we explicitly determine its spectrum, whereas the derivation of the eigenvalues for the second class is left as an open problem. Furthermore, we derive recurrence relations, various identities, and combinatorial sums satisfied by the determinants of matrices of both classes. Another contribution of this work is the formulation of the inverse matrix, the determinant, and the LU-factorization using the determinants of all leading principal Hessenberg submatrices. By specializing the parameters, we establish relationships between these Hessenberg determinants and certain integer or polynomial sequences as well as a factorization of the generalized Fibonacci numbers related to the first class.

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