SOLVABILITY OF A QUATERNION MATRIX EQUATION

This paper discusses the solvability of quaternion matrix equation A *X *B *±B X A=D and obtains it's general explicit solutions in terms of A,B,D and their Moore Penrose inverses.

GH-CONGRUENCE CANONICAL FORM OF QUATERNION MATRIX AND SIMULTANEOUS DIAGONALIZATION OF MATRICES

In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.

Schmidt Decomposition of Quaternion Matrix and the Orthonormalization of Vectors in a Generalized Unitary Space

In this paper we derive a practical method of solving simultaneously the problem of Schmidt decomposition of quaternion matrix and the orthonormalization of vectors in a generalized unitary space by using elementary column operations on matrices over the quaternion field.

Simultaneous diagonalization of two quaternion matrices

The simultaneous diagonalization by congruence of pairs of Hermitian quaternion matrices is discussed. The problem is reduced to a parallel one on complex matrices by using the complex adjoint matrix related to each quaternion matrix. It is proved that any two semi-positive definite Hermitian quaternion matrices can be simultaneously diagonalized by congruence.

QUATERNION GENERALIZED SINGULAR VALUE DECOMPOSITION AND ITS APPLICATIONS

This paper derives a theorem of generalized singular value decomposition of quaternion matrices （QGSVD）,studies the solution of general quaternion matrix equation AXB -CYD= E,and obtains quaternionic Roth＇s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.

ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES

An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.

THE BI-SELF-CONJUGATE AND NONNEGATIVE DEFINITE SOLUTIONS TO THE INVERSE EIGENVALUE PROBLEM OF QUATERNION MATRICES

The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such that AX = X(?), where BSHn×n≥ denotes the set of all n×n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem Ⅱ2= Given B ∈ Hn×m, find B ∈ SE such that ||B-B||Q = minAE∈=sE ||B-A||Q, where SE is the solution set of problem I , || ·||Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.

On the Inequalities for Traces and Singular Values of the Quaternion Matrices

In this paper we derive some inequalities for traces and singular values of the quaternion matrices,extend and improve some of the corresponding results appeared in other papers we know.

The Quaternion Matrix Equation ∑A^iXB_i=E

Let H_F be the generalized quaternion division algebra over a field F with charF≠2. In this paper, the ad joint matrix of any n×n matrix over H_F[λ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this paper obtains several necessary and sufficieut conditions for the existence of a solution or a unique solution to the matrix equation sum from n=i to k A^iXB_i=E over H_F, and gives some explicit formulas of solutions.

Ranks of the Common Solution to Six Quaternion Matrix Equations

A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning （Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195： 721-732 （2008））. Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.

On a Solution of the Quaternion Matrix Equation X-A B=C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.

Least-norm and Extremal Ranks of the Least Square Solution to the Quaternion Matrix Equation AXB = C Subject to Two Equations

In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB = C subject to a consistent system of quaternion matrix equations D1X = F1, XE2 =F2, and derive the maximal and minimal ranks and the leastnorm of the above mentioned solution. The finding of this paper extends some known results in the literature.

A System of Sylvester-type Quaternion Matrix Equations with Ten Variables

This paper studies a system of three Sylvester-type quaternion matrix equations with ten variables A_(i)X_(i)+Y_iB_(i)+C_(i)Z_(i)D_(i)+F_(i)Z_(i+1)G_(i)=E_(i),i=1,3^-.We derive some necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore–Penrose inverses of the matrices involved.We present the general solution to the system when the solvability conditions are satisfied.As applications of this system,we provide some solvability conditions and general solutions to some systems of quaternion matrix equations involvingφ-Hermicity.Moreover,we give some numerical examples to illustrate our results.The findings of this paper extend some known results in the literature.