THE QUATERNION MATRIX EQUATION SIGMA-A(I)XB(I)=E

Let H-F be the generalized quaternion division algebra over a field F with charF not equal 2. In this paper, the adjoint matrix of any n x n matrix over H-F[lambda] is defined and its properties is discussed. B y using the adjoint matrix and the method of representation matrix, th is paper obtains several necessary and sufficient conditions for the e xistence of a solution or a unique solution to the matrix equation Sig ma(i = 0)(k) A(i) x B-i = E over H-F, and gives some explicit formulas of solutions.