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

Emre Kişi,Halim Özdemir,Murat Sarduvan,Nurgül Kalaycı

doi:10.1515/spma-2020-0138

Abstract

Abstract 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.

Highlights

Let N, C, and Cn denote the sets of natural numbers, complex numbers, and n × n complex matrices, respectively. 0 and In will be a zero matrix of appropriate size and the identity matrix of order n, respectively
In [22], Kişi and Özdemir proposed a combinatorial method to handle the problem of characterizing all situations in which the spectrum of the linear combination of nitely many mutually commuting diagonalizable matrices is a subset of a particular set
An algorithm that gives the characterization of the linear combinations of nitely many diagonalizable matrices that mutually commute is proposed

Summary

Introduction

Let N, C, and Cn denote the sets of natural numbers, complex numbers, and n × n complex matrices, respectively. 0 and In will be a zero matrix of appropriate size and the identity matrix of order n, respectively. In [22], Kişi and Özdemir proposed a combinatorial method to handle the problem of characterizing all situations in which the spectrum of the linear combination of nitely many mutually commuting diagonalizable matrices is a subset of a particular set.

Results

Conclusion