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.

ON SOLUTIONS OF QUATERNION MATRIX EQUATIONS XF-AX=BY AND XF-A=BY

In this paper,the quaternion matrix equations XF-AX=BY and XF-A=BY are investigated.For convenience,they were called generalized Sylvesterquaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equation,which include the Sylvester matrix equation and Lyapunov matrix equation as special cases.By applying of Kronecker map and complex representation of a quaternion matrix,the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF-AX=BY and XF-A=BY are also obtained.By the established expressions,it is easy to compute the solution of the quaternion matrix equation in the above two forms.In addition,two practical algorithms for these two quaternion matrix equations are give.One is complex representation matrix method and the other is a direct algorithm by the given expression.Furthermore,two illustrative examples are proposed to show the efficiency of the given method.

6R Robot Inverse Solution Algorithm Based on Quaternion Matrix and Groebner Base

This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

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.

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.

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.

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.

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.

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＋l ＋ 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.

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.

Quaternion-EKF的多源传感器联合定向算法

针对磁力计易受干扰和陀螺仪易漂移的问题,使用扩展卡尔曼滤波EKF(Extended Kalman Filter)融合多源传感器进行定向。利用基于四元数的EKF,在当地磁场和重力加速度的基础上融合加速度计、陀螺仪和磁力计观测信息,并通过自适应算法构建过程噪声向量和观测噪声向量的协方差矩阵以解决载体运动和周围磁场干扰对重力和地磁场观测值的影响。实验结果表明,该算法可以有效地减弱陀螺仪漂移和磁场对定向的干扰,获得更加精确和稳定的航向角信息。

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.

On Bicomplex Representation Methods and Applications of Matrices over Quaternionic Division Algebra

In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express and study. These concepts can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division algebra, JP Journal of Algebra, 2009, 14(2): 121-140] and unify the complex algebra and quaternion division algebra.

GENERALIZED DIAGONALIZATION OF MATRICES OVER QUATERNION FIELD

A concept of [GRAPHICS] diagonalization matrix over quaternion field is given, the necessary and sufficient conditions for determining whether a quaternion matrix is a [GRAPHICS] diagonalization one are discussed, and a method of [GRAPHICS] diagonalization of matrices over quaternion field is given.