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A combinatorial method for determining the spectrum of the linear combinations of finitely many diagonalizable matrices that mutually commute

Emre Kişi
Halim Özdemir


Puni tekst: engleski pdf 220 Kb

str. 61-78

preuzimanja: 464

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Sažetak

et Xi, i=1,2,...,m, be diagonalizable matrices that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix X=i=1mciXi is a matrix such that σ(X){λ1,λ2,...,λn}, where ci, i=1,2,...,m, are nonzero complex scalars and σ(X) denotes the spectrum of the matrix X. If the spectra of the matrices X and Xi, i=1,2,...,m, are chosen as subsets of some particular sets, then this problem is equivalent to the problem of characterizing all situations in which a linear combination of some commuting special types of matrices, e.g. the matrices such that Ak=A, k=2,3,..., is also a special type of matrix. The method developed in this note makes it possible to solve such characterization problems for the linear combinations of finitely many special types of matrices. Moreover, the method is illustrated by considering the problem, which is one of the open problems left in [Linear Algebra Appl. 437 (2012) 2091-2109], of characterizing all situations in which a linear combination X=c1X1+c2X2+c3X3 is a tripotent matrix when X1 is an involutory matrix and both X2 and X3 are tripotent matrices that mutually commute. The results obtained cover those established in the reference above.

Ključne riječi

Diagonalizable matrices; Commutativity; Spectrum; Linear combination; Systems of linear equations

Hrčak ID:

192123

URI

https://hrcak.srce.hr/192123

Datum izdavanja:

30.5.2018.

Posjeta: 1.141 *

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