Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix

The iterative method of generalized coupled Sylvester-transpose linear matrix equations AXB+CYTD=S1,EXTF+GYH=S2 over reflexive or anti-reflexive matrix pair (X,Y)(X,Y) is presented. On the condition that the coupled matrix equations are consistent, we show that the solution pair (X∗,Y∗)(X∗,Y∗) proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff-error for any initial value given a reflexive or anti-reflexive matrix. Moreover, the optimal approximation reflexive or anti-reflexive matrix solution pair to an arbitrary given reflexive or anti-reflexive matrix pair can be derived by searching the least Frobenius norm solution pair of the new generalized coupled Sylvester-transpose linear matrix equations. Finally, some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient.