Research On Solutions To Several Kinds Of Reduced Biquaternion Matrix Equations

Matrix equations play important roles in matrix theory.The theory of reduced biquaternion matrix has an important application in digital image processing,system theory,stability theory,optimal control and neural network.In this paper,let QRBm×n,AHQRBn×n and ABQRBn×n be the m×n reduced biquaternion matrix set,n×n anti-Hermitian matrix set,and n×n anti-bi-Hermitian matrix set,respectively.We mainly study on solving the reduced biquaternion matrix equations,and derive them as the following three problems:Problem I Given A?QRBm×N,B?QBm×p and C?QRm×m,find HE={[X,Y]| X?AHQRBn×n,Y?AHQRBp×p,AXAH+BYBH=C}.Problem ? Given A?QRBm×n,B?QRBn×s,C?QRBm×n,D?QRBn×sand E?QRBm×s,find HE={X|X?AHQRBn×n,AXB+CXD=E}.Problem ? Given A?QRBm×n,B?QRBn×s,C?QRBm×k,D?QRBk×s and E?QRBm×s,find HE={[X,Y]|X?ABQRBn×n,Y?ABQk×k,AXB+CYD=E}.This paper obtains the necessary and sufficient conditions for the solutions of Problems ?-?,derives the expressions for solutions of Problems ?-? when the reduced biquaternion equations are consistent,and provides the numerical algorithm and numerical examples of Problem II to illustrate the feasibility of our algorithm.As conclusions,when these reduced biquaternion equations are inconsistent,we also derive the general expressions for the least squares solution of these equations.