Zhuo-ng He

Solvability conditions and general solution for mixed Sylvester equations

In this paper, we give some necessary and sufficient solvability conditions for the mixed Sylvester matrix equations, and parameterize general solution when it is solvable. Moreover, we investigate the maximal and minimal ranks of the general solution, and maximal and minimal ranks and inertias of Hermitian part of solution, respectively.

A real quaternion matrix equation with applications

Let i, j, k be the quaternion units and let A be a square real quaternion matrix. A is said to be η-Hermitian if −η A*η = A, where η ∈ {i, j, k} and A* is the conjugate transpose of A. Denote A η* = − η A*η. Following Horn and Zhangs recent research on η-Hermitian matrices (A generalization of the complex AutonneTakagi factorization to quaternion matrices, Linear Multilinear Algebra, DOI:10.1080/03081087.2011.618838), we consider a real quaternion matrix equation involving η-Hermicity, i.e. where Y and Z are required to be η-Hermitian. We provide some necessary and sufficient conditions for the existence of a solution (X, Y, Z) to the equation and present a general solution when the equation is solvable. We also study the minimal ranks of Y and Z satisfying the above equation.

Some matrix equations with applications

We establish necessary and sufficient conditions for the solvability to the matrix equation and present an expression of the general solution to (1) when it is solvable. As applications, we discuss the consistence of the matrix equation where * means conjugate transpose, and provide an explicit expression of the general solution to (2). We also study the extremal ranks of X 3 and X 4 and extremal inertias of and in (1). In addition, we obtain necessary and sufficient conditions for the classical matrix equation to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite and Re-negative definite solutions. The findings of this article extend related known results. †Dedicated to Professor Ky Fan (1914–2010).

Systems of coupled generalized Sylvester matrix equations

This paper studies some systems of coupled generalized Sylvester matrix equations. We present some necessary and sufficient conditions for the solvability to these systems. We give the expressions of the general solutions to the systems when their solvability conditions are satisfied.

The general solutions to some systems of matrix equations

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 systemwhere means conjugate transpose. We also give the expressions of the general solutions to the systems.

Constraint generalized Sylvester matrix equations

In this paper, some necessary and sufficient conditions are established for the constraint generalized Sylvester matrix equations to have a common solution. The expression of the general common solution is also given under the solvable conditions. In addition, a numerical example is presented to illustrate the presented theory.

The η-bihermitian solution to a system of real quaternion matrix equations

Let be the set of all matrices over the real quaternion algebra. , , , where is the conjugate of the quaternion . We call that is -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.

Solution to a system of real quaternion matrix equations encompassing η-Hermicity

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 ?-Hermitian if A = A ? * = - ? A * ? , ? ? { 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 ?-Hermicity A 1 X = C 1 , A 2 Y = C 2 , Y B 2 = D 2 , Y = Y ? * , A 3 Z = C 3 , Z B 3 = D 3 , Z = Z ? * , A 4 X + ( A 4 X ) ? * + B 4 Y B 4 ? * + C 4 Z C 4 ? * = 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.

Simultaneous decomposition of quaternion matrices involving ź-Hermicity with applications

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 ź R } . For ź ź {i, j, k}, a real quaternion matrix A ź H n × n is said to be ź-Hermitian if A ź * = A where A ź * = - ź A * ź , 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 ź-Hermicity with compatible sizes: A i ź H p i × t i , B i ź H p i × t i + 1 , and C i ź H p i × p i , where Ci are ź-Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an ź-Hermitian solution to the system of coupled real quaternion matrix equations A i X i A i ź * + B i X i + 1 B i ź * = C i , ( i = 1 , 2 , 3 ) , and provide an expression of the general ź-Hermitian solutions to this system. Moreover, we derive the rank bounds of the general ź-Hermitian solutions to the above-mentioned system using ranks of the given matrices Ai, Bi, and Ci as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper.

A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+1 + YkDk = Nk (k = 1, 2) over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.