Generally Unitary Solution to a System of Matrix Equations over the Quaternion Field

Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence of and the expression for the generally unitary solution of the system are derived.

Minor Self-conjugate and Skewpositive Semidefinite Solutions to a System of Matrix Equations over Skew Fields

Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.

The General and Centro(Kewsy) Symmetric Solutions to a System of Matrix Equations over an Arbitrary Skew Field

Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.

On a Matrix Equation AX+XB=C over a Skew Field

In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.

The Matrix Equation A_(mn)X_(ns)B_(st)=Cover an Arbitrary Skew Field

The matrix equation A mn X ns B st =C over an arbitrary skew field is considered. A necessary and sufficient condition for the consistency and the expression for general solutions of the above mentioned matrix equation are presented.Moreover,a practical method of solving one is also given.

A Necessary and Sufficient Condition for Solvability of the Matrix Equation AXB＋CYD＝E Over an Arbitrary Skew Field

A necessary and sufficient condition for the consisteney of the matrix equation AXB+CYD=E over an arbitrary skew field is given. Thus Roth's equivalence theorem is generalized.

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.

The Least Square Solutions to the Quaternion Matrix Equation AX=B

A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

Left Eigenvector of a Stochastic Matrix

We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .

Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation

In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .

体上某些分块矩阵的Drazin逆(英文)

令?n 记 体?上 的 所 有n × n矩 阵 的 集 合. 对 于 一 个 固 定 的A∈?n , 若 正 整 数k = ×n ×nmin{l|Al X = Al 对某个X ∈ ?n },则称k为A的指标. 如果X ∈ ?n 满足下面的方程组AX = +1 ×n ×nXA,X2A = X,Ak X = Ak,其中k为A的指标,则称X为A的Drazin逆,当k=1时,A# = AD被称 +1为A的群逆. ?n 的某些分块矩阵的Drazin逆和群逆的存在性和表示被给出.

四元数体上矩阵方程AXA*=B的非负定解(英文)

讨论了四元数体上矩阵方程AXA*=B的非负定解,解决了以下问题:(1)给出了四元数体上矩阵方程AXA*=B存在非负定解的充分必要条件;(2)当矩阵方程AXA*=B的非负定解存在时,给出了通解的表达式和解的加法分解式;(3)若X是矩阵方程AXA*=B的非负定解,给出了求X的秩的公式以及X为最小秩或最大秩解的条件.

四元数体上矩阵方程{AXA^＊=B CXC^＊=D的非负定解

本文讨论四元数体上矩阵方程AXA*=BCXC*=D的非负定解,解决了以下问题:(1)给出了矩阵方程AXA*=BCXC*=D存在非负定解的充分必要条件;(2)当矩阵方程AXA*=BCXC*=D有非负定解时,给出了通解的表达式;(3)当矩阵方程AXA*=BCXC*=D有非负定解X时,给出了X的秩的范围.

四元数体上统一代数Lyapunov方程的循环解及最佳逼近

运用四元数矩阵的复表示运算和矩阵的Kronecker积,并结合循环矩阵的特殊结构,获得了四元数体上统一代数Lyapunov方程具有循环解的充要条件及其解的一般表达式.在循环解集中得到预先给定的四元数循环矩阵有极小Frobenius范数的最佳逼近解.

体上矩阵方程AXB＝C的求解

给出了任意体上的矩阵方程ＡＸＢ＝Ｃ相容的充要条件及其通解的矩阵算法。

任意体上的矩阵方程AXB+CYD=O

运用任意体上矩阵的广义逆，给出了任意体上矩阵方程ＡＸＢ＋ＣＹＤ＝Ｏ的通解表达式及其仅有零解的一个充要条件．

关于四元数矩阵方程AXA^＊＝B的最小二乘解

定义了四元数矩阵方程的范数 ,导出了四元数矩阵方程AXA =B的最小二乘解及其在约束条件DX =E 下的最小二乘解 ,以及其具有极小范数的最小二乘解 .

Rellman猜想在四元数体的修正

本文就ｎ＝２￣ｔ（ｔ为自然数）时，证明了不等式：Ｒｅ［ｔｒ（ＡＢ）￣ｎ］≤Ｒｅ［ｔｒＡ￣ｎＢ￣ｎ］成立，其中Ａ，Ｂ为半正定四元数四共轭矩阵。

关于任意体上的矩阵方程X_（m×n）A_（n×s）＝Y_（m×t）B_（t×s

关于四元数体上的矩阵方程已有不少研究，但对任意体上矩阵方程的研究还很不深入。本文运用矩阵技巧及含幺环上矩阵的初等变换对任意体上的矩阵方程进行了研究，给出了此类方程的通解表达式及其实用解法。