A MODIFIED HSS ITERATION METHOD FOR SOLVING THE COMPLEX LINEAR MATRIX EQUATION AXB = C

In this paper, a modified Hermitian and skew-Hermitian splitting （MHSS） iteration method for solving the complex linear matrix equation AXB = C has been presented. As the theoretical analysis shows, the MHSS iteration method will converge under cer- tain conditions. Each iteration in this method requires the solution of four linear matrix equations with real symmetric positive definite coefficient matrices, although the original coefficient matrices are complex and non-Hermitian. In addition, the optimal parameter of the new iteration method is proposed. Numerical results show that MHSS iteration method is efficient and robust.

ON DETERMINATION OF THE IMPULSIVE SOLUTIONS AND IMPULSIVE PROPERTIES TO LINEAR NON-HOMOGENEOUS MATRIX DIFFERENTIAL EQUATIONS

This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.

Parameterized Solution to a Class of Sylvester Matrix Equations

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.

Closed-form Solutions to the Matrix Equation AX-EXF=BY with F in Companion Form

A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .

A system of matrix equations and its applications

We derive the solvability conditions and an expression of the general solution to the system of matrix equations A 1X=C1 , A2Y=C2 , YB2=D2 , Y=Y*, A3Z=C3 , ZB3=D3 , Z=Z*, B4X+(B4X)+C4YC4*+D4ZD4*=A4 . Moreover, we investigate the maximal and minimal ranks and inertias of Y and Z in the above system of matrix equations. As a special case of the results, we solve the problem proposed in Farid, Moslehian, Wang and Wu's recent paper (Farid F O, Moslehian M S, Wang Q W, et al. On the Hermitian solutions to a system of adjointable operator equations. Linear Algebra Appl, 2012, 437: 1854-1891).

An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications

We in this paper give a decomposition concerning the general matrix triplet over an arbitrary divisionring F with the same row or column numbers. We also design a practical algorithm for the decomposition of thematrix triplet. As applications, we present necessary and suficient conditions for the existence of the generalsolutions to the system of matrix equations DXA = C1, EXB = C2, F XC = C3 and the matrix equation AXD + BY E + CZF = Gover F. We give the expressions of the general solutions to the system and the matrix equation when thesolvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of thispaper. We also mention the other applications of the equivalence canonical form, for instance, for the compressionof color images.