373 research outputs found
The Drazin inverse of the linear combinations of two idempotents in the Banach algebra
AbstractIn this paper, some Drazin inverse representations of the linear combinations of two idempotents in a Banach algebra are obtained. Moreover, we present counter-examples to and establish the corrected versions of two theorems by Cvetković-Ilić and Deng
Idempotency of linear combinations of three idempotent matrices, two of which are disjoint
AbstractGiven nonzero idempotent matrices A1,A2,A3 such that A2 and A3 are disjoint, i.e., A2A3=0=A3A2, the problem of characterizing all situations, in which a linear combination C=c1A1+c2A2+c3A3 is an idempotent matrix, is studied. The results obtained cover those established by J.K. Baksalary, O.M. Baksalary, and G.P.H. Styan (Linear Algebra Appl. 354 (2002) 21) under the additional assumption that c3=−c2, i.e., in the particular case where C=c1A1+c2(A2−A3) is actually a linear combination of an idempotent matrix A1 and a tripotent matrix A2−A3
A combinatorial method for determining the spectrum of the linear combinations of finitely many diagonalizable matrices that mutually commute
et , , be diagonalizable matrices that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix is a matrix such that , where , , are nonzero complex scalars and denotes the spectrum of the matrix . If the spectra of the matrices and , , 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 , , 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 is a tripotent matrix when is an involutory matrix and both and are tripotent matrices that mutually commute. The results obtained cover those established in the reference above
A combinatorial method for determining the spectrum of the linear combinations of finitely many diagonalizable matrices that mutually commute
et , , be diagonalizable matrices that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix is a matrix such that , where , , are nonzero complex scalars and denotes the spectrum of the matrix . If the spectra of the matrices and , , 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 , , 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 is a tripotent matrix when is an involutory matrix and both and are tripotent matrices that mutually commute. The results obtained cover those established in the reference above
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