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We in this paper define the outer-Perron-Frobenius splitting, which is an extension of the pseudo- Perron-Frobenius splitting defined in [A.N. Sushama, K. Premakumari, K.C. Sivakumar, Extensions of Perron-Frobenius splittings and relationships with nonnegative Moore-Penrose inverse, Linear and Multilinear Algebra 63 (2015) 1-11]. We present some criteria for the convergence of the outer-Perron-Frobenius splitting. The findings of this paper generalize some known results in the literatures.

Although solvability conditions for a system of two linear equations are well-known even in the case of rings and, for three linear equations, in the case of matrices, in the case of four linear equations there are no results. In this paper, we consider systems of four linear matrix equations A i X B i = C i , i = 1 , 4 ¯ and present some necessary and sufficient conditions for their solvability as well as an expression for the general solution. There are two advantages to our results: the presented solvability conditions in many cases can be presented in a purely algebraic form and the method used in the proof allows for a generalization of the obtained results to some more general structures such as algebras of bounded linear operators or rings, under some additional assumptions concerning regularity only. We present several applications of our results. [ABSTRACT FROM AUTHOR]

In this paper, the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse, the e-core inverse and the f-dual core inverse in rings. Also, new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given. [ABSTRACT FROM AUTHOR]

Let R be a unital ∗ -ring and let a , e , f ∈ R with e,f invertible Hermitian elements. In this paper, we define two types of outer generalized inverses, called pseudo e-core inverses and pseudo f-dual core inverses. An element a ∈ R is pseudo e-core invertible if there exist an element x ∈ R and some positive integer n such that x a x = x , x R = a n R and R x = R (a n) ∗ e. Dually, a is pseudo f-dual core invertible if there exist an element x ∈ R and some positive integer m such that xax=x, R x = R a m and f x R = (a m) ∗ R. Moreover, we investigate both of them for their characterizations and properties. Also, the relations between the pseudo e-core inverse (resp. the pseudo f-dual core inverse) and the inverse along an element are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equations A 1 X B 1 = C 1 , A 2 X B 2 = C 2 , and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equation A X B + C Y D = E , respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature. [ABSTRACT FROM AUTHOR]

Let R and H m × n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebra H = { a 0 + a 1 i + a 2 j + a 3 k | i 2 = j 2 = k 2 = ijk = − 1 , a 0 , a 1 , a 2 , a 3 ∈ R } . For η ∈ { i, j, k }, a real quaternion matrix A ∈ H n × n is said to be η -Hermitian if A η * = A where A η * = − η A * η , and A * stands for the conjugate transpose of A , arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η -Hermicity with compatible sizes: A i ∈ H p i × t i , B i ∈ H p i × t i + 1 , and C i ∈ H p i × p i , where C i are η -Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η -Hermitian solution to the system of coupled real quaternion matrix equations A i X i A i η * + B i X i + 1 B i η * = C i , ( i = 1 , 2 , 3 ) , and provide an expression of the general η -Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η -Hermitian solutions to the above-mentioned system using ranks of the given matrices A i , B i , and C i as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

We provide two techniques, which establish criteria for the consistency of a system of matrix equations (see (1.1)). The system encompasses matrix systems that were not studied before. We present the solutions set of a consistent system (1.1), using each technique. We investigate the link between the two techniques. We study the number of solutions that a system (1.1) could have, and establish a necessary and sufficient condition for a consistent system (1.1) to have a unique solution. If,and,, are all the zero matrices in (1.1) and the system is consistent, we provide bounds for the rank and inertia of any Hermitian solution of the system. [ABSTRACT FROM PUBLISHER]

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations A 1 X = C 1 , Y B 1 = D 1 , A 2 W = C 2 , Z B 2 = D 2 , A 3 V = C 3 , V B 3 = C 4 , A 4 V B 4 = C 5 , A 5 X + Y B 5 + C 6 W + Z D 6 + E 6 V F 6 = G 6 , and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider an expression of the general solution to the classical system of matrix equationsWe present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the systemand the systemwheremeans conjugate transpose. We also give the expressions of the general solutions to the systems. [ABSTRACT FROM AUTHOR]

Let H m × n be the set of all m × n matrices over the real quaternion algebra H = { c 0 + c 1 i + c 2 j + c 3 k ∣ i 2 = j 2 = k 2 = i j k = − 1 , c 0 , c 1 , c 2 , c 3 ∈ R } . A ∈ H n × n is known to be η -Hermitian if A = A η * = − η A * η , η ∈ { i , j , k } and A * means the conjugate transpose of A . We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η -Hermicity A 1 X = C 1 , A 2 Y = C 2 , Y B 2 = D 2 , Y = Y η * , A 3 Z = C 3 , Z B 3 = D 3 , Z = Z η * , A 4 X + ( A 4 X ) η * + B 4 Y B 4 η * + C 4 Z C 4 η * = D 4 , and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise. [ABSTRACT FROM AUTHOR]

In this paper, we define and discuss the Moore–Penrose inverses of matrices with quaternion polynomial entries. When the Moore–Penrose inverses exist, we prove that Leverrier–Faddeev algorithm works for these matrices by using generalized characteristic polynomials. Furthermore, after studying interpolations for quaternion polynomials, we give an efficient algorithm to compute the Moore–Penrose inverses. We developed a Maple package for quaternion polynomial matrices. All algorithms in this paper are implemented, and tested on examples. [ABSTRACT FROM AUTHOR]

Letbe the set of allmatrices over the real quaternion algebra.,,, whereis the conjugate of the quaternion. We call thatis-Hermitian, if,;is-bihermitian, if. We in this paper, present the solvability conditions and the general-Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an-bihermitian solution. We establish an expression of the-bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be-bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper. [ABSTRACT FROM PUBLISHER]

For the general linear mixed model ℱ: {y,Xβ + Zu + e,Σ}, where the fixed effectsuand random effectseare correlated and the covariance matrixΣmay be singular. We establish a new explicit expression of the best linear unbiased predictors (BLUPs) and give the necessary and sufficient conditions for the BLUP of the predictable parameter functionLβ + Muunder the ℱ to remain BLUP under its misspecified model {y,X0β + Z0u + e,Σ0}. We also derive the necessary and sufficient conditions for the BLUPs under the multiple partitioned linear mixed model {y,X1β1 + X2β2+ … +Xkβk + Z1u1 + Z2u2+ … +Zkuk + e,Σ} to be the sum of BLUPs under its small models {y,Xiβi + Ziui + e,Σi},i = 1, 2,…,k. [ABSTRACT FROM PUBLISHER]

In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A1 XB1 = C1, A2XB2 = C2. Moreover, formulas of the maximal and minimal ranks of four real matrices X1, X2, X3, and X4 in solution X = X1 + X2i + X3j + X4k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A1XB1 = C1, A2XB2 = C2, A3XB3 = C3 to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A1 + B1i + C1j + D1k and A2 + B2i + C2j + D2k, which can be expressed as E + Fi + Gj + Hk are also presented. [ABSTRACT FROM AUTHOR]