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

Abstract The iterative method of generalized coupled Sylvester-transpose linear matrix equations A X B + C Y T D = S 1 , E X T F + G Y H = 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.

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