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It is established that the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation \({(AXB,DXE)=(C,F)}\) for centrohermitian and skew-centrohermitian matrices, where X is an unknown quaternion matrix and A, B, C, D, E, and F are known quaternion matrices of suitable sizes. Moreover, an algorithm to get the solutions of the problems considered is provided, and a numerical example is also given to exemplify the results.

Let A(1) and A(2) be two nonzero commuling} and {a2, fi2}-quadratic matrices, respectively, where alpha(1), beta(1), alpha(2), beta(2) is an element of C with alpha(1) not equal beta(1) and alpha(2) not equal beta(2). The aim of this work is mainly to characterize all situations, where the linear combination a(1)A(1) + a(2)A(2) is a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, in volutivi ty and tripotency of the linear combinations of idempotent and/or in volutive matrices.

Let and be nonzero quadratic matrices and let and be nonzero complex numbers. Necessary and sufficient conditions for the quadraticity of the linear combinations of the form are obtained. Our main results contain many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices. Finally, some numerical examples are given to exemplify the main results.

Suppose that the matrix equations system (A(1)XB(1), ... , A(k)XB(k)) = (C-1,..., C-k) with unknown matrix X is given, where A(i), B-i, and C-i, i = 1, 2,..., k, are known matrices of suitable sizes. The matrix nearness problem is considered over the general and least squares solutions of this system. The explicit forms of the best approximate solutions of the problems over the sets of symmetric and skew-symmetric matrices are established as well. Moreover, a comparative table depending on some numerical examples in the literature is given. Research Fund of the Sakarya UniversitySakarya University [2014-02-00-001] This work was supported by Research Fund of the Sakarya University. Project Number: 2014-02-00-001.

Let T 1 and T 2 be two nonzero commuting n × n tripotent matrices and c 1 , c 2 two nonzero complex numbers. Necessary and sufficient conditions for the tripotency and the idempotency of c 1 T 1 + c 2 T 2 are obtained. The problems considered here have also statistical importance when c 1 , c 2 are real scalars and T 1 , T 2 are real symmetric matrices.

Let A = c 1 A 1 + c 2 A 2 , where c 1 , c 2 are nonzero complex numbers and ( A 1 , A 2 ) is a pair of two n × n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A = c 1 A 1 + c 2 A 2 is (i) a tripotent or an involutive matrix when A 1 and A 2 are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when A 1 and A 2 are involutive matrices, and (iii) an involutive matrix when A 1 and A 2 are involutive or idempotent matrices.

It is investigated the necessary and sufficient conditions for the generalized quadraticity of a linear combination of any two generalized quadratic matrices. The main result obtained is, in a sense, a generalization of the main results given in [Uc M, Ozdemir H, Ozban AY. On the quadraticity of linear combinations of quadratic matrices. Linear Multilinear Algebra. 2015;63:1125–1137.] which contains many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices, to the generalized quadratic matrices.

Let A and B be two generalized quadratic matrices with respect to idempotent matrices P and Q, respectively, such that (A - alpha P)(A - beta P) = 0, AP = PA = A, 113 y (2)(B - gamma Q) (B - delta Q) = 0, BQ = QB - B PQ - QP, AB not equal BA, and (A + B)(alpha beta P - gamma delta Q) - (alpha beta P - gamma delta Q)(A + B) with alpha, beta, gamma, delta is an element of C. Let A + B be diagonalizable. The relations between the spectrum of the matrix A + B and the spectra of some matrices produced from A and B are considered. Moreover, some results on the spectrum of the matrix A + B are obtained when A B is not diagonalizable. Finally, some results and examples illustrating the applications of the results in the work are given. (C) 2015 Elsevier Inc. All rights reserved.

P-1, P-2 and P-3 being any three different nonzero mutually commutative n x n idempotent matrices, and C-1 C-2 and C-3 being nonzero scalars, the problem of characterizing some situations, where a linear combination of the form P = c(1)P(1) + c(2)P(2) or P = c(1)P(1) + c(2)P(2) + c(3)P(3), is also an idempotent matrix has been considered. Moreover two interesting results about the idempotency of linear combinations of 2 x 2 idempotent matrices and 3 x 3 idempotent matrices have been given. A statistical interpretation of the idempotency problem considered in this study has also been pointed out. (C) 2003 Elsevier Inc. All rights reserved.

The purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix. In the case of real scalars and real symmetric matrices, this problem admits an interesting statistical interpretation. Namely, it is equivalent to the question of when a linear combination of two quadratic forms in normal variables, each distributed as a difference of two independent chi(2)-variables, is also distributed as such a difference. (C) 2004 Published by Elsevier Inc.

The relations between the spectrum of the matrix Q + R and the spectra of matrices (gamma + delta)Q + (alpha+ beta)R - QR - RQ,QR - RQ, alpha beta R - QRQ, alpha RQR - (QR)(2), and beta R - QR have been given assuming that the matrix Q + R is diagonalizable, where Q, R are {alpha, beta} -quadratic matrix and {gamma, delta}-quadratic matrix, respectively, of order n.

Let T=c1T1+c2T2+c3T3- c4(T1T2+T3T1+T2T3), where T1, T2, T3 are three n x n tripotent matrices and c1, c2, c3, c4 are complex numbers with c1, c2, c3 nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices., We would like to thank the referee for his/her careful reading. The first author was supported by the Vicerrectorado de Investigacion U.P.V. PAID 06-2010-2285.

It has been established a 3(n)-term disjoint idempotent decomposition ( DID) for the linear combinations produced from n (>= 2) commutative tripotent matrices, their products and their products of power 2 at most. The results obtained in this way generalize those in [Y. Tian, A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications, Linear Multilinear Algebra 59 (2011) 1237-1246]. Moreover, an algorithm to get a DID has been provided. Finally, a numerical example has been given to exemplify the results. (C) 2012 Elsevier Inc. All rights reserved.