Qing-Wen Wang

Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra

Let Ω be a finite dimensional central algebra with an involutorial antiautomorphism σ and char Ω≠2, Ωn×n be the set of all n×n matrices over Ω. A=(aij)∈Ωn×n is called bisymmetric if aij=an−i+1,n−j+1=σ(aji) and biskewsymmetric if aij=−an−i+1,n−j+1=−σ(aji). The following systems of generalized Sylvester equations over Ω[λ]: A1X−YB1=C1, (I)⋮ AsX−YBs=Cs, A1XB1−C1XD1=E1, (II)⋮ AsXBs−CsXDs=Es, are considered. Necessary and sufficient conditions are given for the existence of constant solutions with bi(skew)symmetric constrains to (I) and (II). As a particular case, auxiliary results dealing with the system of Sylvester equations are also presented.

THE REFLEXIVE RE-NONNEGATIVE DEFINITE SOLUTION TO A QUATERNION MATRIX EQUATION ∗

In this paper a necessary and sufficient condition is established for the existence of the reflexive re-nonnegative definite solution to the quaternion matrix equation AXA ∗ = B ,w here ∗ stands for conjugate transpose. The expression of such solution to the matrix equation is also given. Furthermore, a necessary and sufficient condition is derived for the existence of the general re-nonnegative definite solution to the quaternion matrix equation A1X1A ∗ + A2X2A ∗ = B.T he representation of such solution to the matrix equation is given.

ON SOLUTIONS TO THE QUATERNION MATRIX EQUATION AXB + CY D = E ∗

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi,Yi,i =1 ,···, 4, in a solution pair X = X1 +X2i +X3j +X4k and Y = Y1 +Y2i +Y3j +Y4k to this equation. Moreover, necessary and sufficient conditions are derived for all solution pairs X and Y of this equation to be real or pure imaginary, respectively. Some known results can be regarded as special cases of the results in this paper.

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).

Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications

Abstract In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 - A 4 XB 4 where X is a variant quaternion matrix subject to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 , A 4 XB 4 = C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D - CA - B with respect to an inner inverse A − of A , which is a common solution to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper.

Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application

Abstract In this paper, we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations A 1 X = C 1 , A 2 X = C 2 , A 3 XB 3 = C 3 and A 4 XB 4 = C 4 , which was investigated recently by Wang [Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665–675]. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented. Some previous known results can be regarded as the special cases of this paper.

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Geršgorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauers theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Geršgorin.

A new solvable condition for a pair of generalized Sylvester equations

A necessary and sufficient condition is given for the quaternion matrix equations AiX + YB i = Ci (i =1 , 2) to have a pair of common solutions X and Y. As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + YB = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

EXTREME RANKS OF (SKEW-)HERMITIAN SOLUTIONS TO A QUATERNION MATRIX EQUATION ¤

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA ¤ + BY B ¤ = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1,··· , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y = Y1 + Y2i + Y3j + Y4k. Moreover, the necessary and sufficient conditions for the existence of a real (skew-)symmetric solution, a complex (skew-)Hermitian solution, and a pure imaginary (skew-)Hermitian solution to the matrix equation mentioned above are presented in this paper. Also established are expressions of such solutions to the equation when corresponding solvability conditions are satisfied. The findings of this paper widely extend the known results in the literature.