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

摘要

The iterative method of generalized coupled Sylvester-transpose linear matrix equations AXB + CY~TD = S_1, EX~TF + GYH = S_2 over reflexive or anti-reflexive matrix pair (X, Y) is presented. On the condition that the coupled matrix equations are consistent, we show that the solution pair (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.