On linear combinations of generalized involutive matrices

We propose an algorithm, which is based on the method given by Kişi and Özdemir in [Math Commun, 23 (2018) 61], to handle the problem of when a linear combination matrix X = ∑ i = 1 m c i X i X = \sum\nolimits_{i = 1}^m {{c_i}{X_i}} is a matrix such that its spectrum is a subset of a particular set, where ci , i = 1, 2, ..., m, are nonzero scalars and Xi , i = 1, 2, ..., m, are mutually commuting diagonalizable matrices. Besides, Mathematica implementation codes of the algorithm are also provided. The problems of characterizing all situations in which a linear combination of some special matrices, e.g. the matrices that coincide with some of their powers, is also a special matrix can easily be solved via the algorithm by choosing of the spectra of the matrices X and Xi , i = 1, 2, ..., m, as subsets of some particular sets. Nine of the open problems in the literature are solved by utilizing the algorithm. The results of the four of them, i.e. cubicity of linear combinations of two commuting cubic matrices, quadripotency of linear combinations of two commuting quadripotent matrices, tripotency of linear combinations of three mutually commuting tripotent matrices, and tripotency of linear combinations of four mutually commuting involutive matrices, are presented explicitly in this work. Due to the length of their presentations, the results of the five of them, i.e. quadraticity of linear combinations of three or four mutually commuting quadratic matrices, cubicity of linear combinations of three mutually commuting cubic matrices, quadripotency of linear combinations of three mutually commuting quadripotent matrices, and tripotency of linear combinations of four mutually commuting tripotent matrices, are given as program outputs only. The results obtained are extensions and/or generalizations of some of the results in the literature.

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On the spectrum of linear combinations of finitely many diagonalizable matrices that mutually commute

Kişi

Sarduvan

Özdemir

et al. 2021

Special Matrices

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Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

Yao

Zheng

2012

ISRN Applied Mathematics

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Let X and Y be infinite dimensional Banach spaces over the real or complex field 𝔽, and let 𝒜 and ℬ be standard operator algebras on X and Y, respectively. In this paper, the structures of surjective maps from 𝒜 onto ℬ that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.

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On linear combinations of generalized involutive matrices

Abstract: Let X † denotes the Moore-Penrose pseudoinverse of a matrix X. We study a number of situations when (aA + bB) † = aA + bB provided a, b ∈ C \ {0} and A, B are n × n complex matrices such that A † = A and B † = B.

Cited by 2 publications

References 14 publications

On the spectrum of linear combinations of finitely many diagonalizable matrices that mutually commute

On the spectrum of linear combinations of finitely many diagonalizable matrices that mutually commute

Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

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