Study On Solutions For Several Classes Of Matrix Difference Equation

Difference equations are suitable for explanation and discussion to many practical problems with discrete variable. It is widely applicable in many aspects such as numerical calculation, combinated count, linear system analysis, etc. In the field of numerical calculation, it is an important method of numerically solving a large quantity of differential equations, to change continuous variables into discrete ones and to change continuously differential equations into discretely difference equations. For difference equations consisted of functions with integer scalar quantity, using matrix form to express is especially convenient and easy to discuss its solving method.In this paper, we mainly investigate the general solutions for several classes of matrix difference equation and their asymptotic stability. The concrete content is as follows:(1) Using iteration, the general solution and its asymptotic stability of one order matrix difference equation X_n+1=AX_n,and the general solution of X_n+1=AX_n+Ris investigated (where A , B , X_n∈R^m×m,{X_n}is a sequence of matrix functions with integer scalar quantity).(2) Using quadratic characteristic matrix and its characteristic polynomial, the general solution and its asymptotic stability of second order homogenous matrix difference equation X_n+2-BX_n+1-AX_n=0 is investigated (whereA , B , X_n∈R^m×m,｛X_n｝is a sequence of matrix functions with integer scalar quantity). Xn}(3) Using k -th characteristic matrix and its characteristic polynomial, the general solution and its asymptotic stability of k -order homogenous matrix difference equationX n ? k? X_n+k+A_k-1X_n+k-1+…+A₁X_n+1+A_OX_n=O is investigated (where,A_i(i=0,1,…,k-1),X_n∈C^m×m{X_n}is a sequence of matrix functions with integer scalar quantity).(4) Based on Jordan decomposition of a matrix and companion matrix, the solution method of the following n -th matrix equation A_nⁿ+A_n-1T^n-1+…+A₁T+A₀=0 is discussed. where A₀,A₁,…,A_n∈C^m×m are known,T∈C^m×m is unknown.(5) Discuss the general solution of two variables matrix difference equation below: X(t,s)=BX(t,s-1)+CX(t-1,s)+DX(t-1,s-1)+E(t,s)where X (t , s )is unknown sequence of matrix functions with two variables t ,s, B , C , D∈R^n×n are constant matrices, and E (t , s)is known matrix functions with two variables.