Algorithms for simultaneous block triangularization and block diagonalization of sets of matrices

Ahmad Y. Al-Dweik; Ryad Ghanam; Gerard Thompson; M. T. Mustafa; Ahmad Y. Al-Dweik; Ryad Ghanam; Gerard Thompson; M. T. Mustafa

doi:10.3934/math.20231007

AIMS Mathematics

2023, Volume 8, Issue 8: 19757-19772. doi: 10.3934/math.20231007

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Research article Special Issues

Algorithms for simultaneous block triangularization and block diagonalization of sets of matrices

1.
Department of Mathematics, Birzeit University, Ramallah, Palestine
2.
Department of Liberal Arts & Sciences, Virginia Commonwealth University in Qatar, Doha 8095, Qatar
3.
Department of Mathematics, University of Toledo, Toledo, OH 43606, USA
4.
Mathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, 2713, Doha, Qatar

Received: 23 April 2023 Revised: 16 May 2023 Accepted: 18 May 2023 Published: 13 June 2023
MSC : 15A75, 47A15, 68-04

In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper investigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigenvectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.
- invariant subspace,
- block-triangular form,
- block-diagonal form,
- composition series
Citation: Ahmad Y. Al-Dweik, Ryad Ghanam, Gerard Thompson, M. T. Mustafa. Algorithms for simultaneous block triangularization and block diagonalization of sets of matrices[J]. AIMS Mathematics, 2023, 8(8): 19757-19772. doi: 10.3934/math.20231007

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Abstract

In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper investigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigenvectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.

References

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