373 research outputs found

    The Drazin inverse of the linear combinations of two idempotents in the Banach algebra

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    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

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    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

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    et XiX_i, i=1,2,...,mi=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=sumi=1mciXiX=sum_{i=1}^{m}c_iX_i is a matrix such that sigma(X)subseteqlambda1,lambda2,...,lambdansigma(X)subseteq{lambda_1, lambda_2,..., lambda_{n}}, where cic_i, i=1,2,...,mi=1,2,...,m, are nonzero complex scalars and sigma(X)sigma(X) denotes the spectrum of the matrix XX. If the spectra of the matrices XX and XiX_i, i=1,2,...,mi=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=AA^k=A, k=2,3,...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+c3X3X=c_1X_1+c_2X_2+c_3X_3 is a tripotent matrix when X1X_1 is an involutory matrix and both X2X_2 and X3X_3 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

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    et XiX_i, i=1,2,...,mi=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=sumi=1mciXiX=sum_{i=1}^{m}c_iX_i is a matrix such that sigma(X)subseteqlambda1,lambda2,...,lambdansigma(X)subseteq{lambda_1, lambda_2,..., lambda_{n}}, where cic_i, i=1,2,...,mi=1,2,...,m, are nonzero complex scalars and sigma(X)sigma(X) denotes the spectrum of the matrix XX. If the spectra of the matrices XX and XiX_i, i=1,2,...,mi=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=AA^k=A, k=2,3,...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+c3X3X=c_1X_1+c_2X_2+c_3X_3 is a tripotent matrix when X1X_1 is an involutory matrix and both X2X_2 and X3X_3 are tripotent matrices that mutually commute. The results obtained cover those established in the reference above
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