GENERALIZED LYAPUNOV EQUATIONS, MATRICES WITH DISPLACEMENT STRUCTURE,AND GENERALIZED BEZOUTIANS

Matrix equations of the form SIGMA(i=0)(r)SIGMA(j=0)(s)f(ij)A(i)XB(j) = C are considered. It is shown that solving such equations can be red uced to finding the solutions of certain well-chosen auxiliary equatio ns. The latter equations are essentially simpler than the original equ ation, and their solutions are matrices with a displacement structure or connected with generalized Bezoutians. In the case r = s = 1, the a pproaches given here, along with some results on structured matrices a nd generalized Bezoutians, lead to explicit formulas for the solution.