Abstract
In this paper, the generalized Sylvester matrix equation AV + BW = EVF + C over reflexive matrices is considered. An iterative algorithm for obtaining reflexive solutions of this matrix equation is introduced. When this matrix equation is consistent over reflexive solutions then for any initial reflexive matrix, the solution can be obtained within finite iteration steps. Furthermore, the complexity and the convergence analysis for the proposed algorithm are given. The least Frobenius norm reflexive solutions can also be obtained when special initial reflexive matrices are chosen. Finally, numerical examples are given to illustrate the effectiveness of the proposed algorithm.
Highlights
Introduction and preliminaries Consider the generalizedSylvester matrix equationAV þ BW 1⁄4 EVF þ C; ð1:1Þ where A, E ∈ Rm × p, B ∈ Rm × q, F ∈ Rn × n, and C ∈ Rm × n while V ∈ Rp × n and W ∈ Rq × n are matrices to be determined
An n × n real matrix P ∈ Rn × n is called a generalized reflection matrix if PT = P and P2 = I
We define the inner product of two matrices A, B as 〈A, B〉 = tr(BTA)
Summary
Introduction and preliminaries Consider the generalizedSylvester matrix equationAV þ BW 1⁄4 EVF þ C; ð1:1Þ where A, E ∈ Rm × p, B ∈ Rm × q, F ∈ Rn × n, and C ∈ Rm × n while V ∈ Rp × n and W ∈ Rq × n are matrices to be determined. Ramadan et al [8] considered explicit and iterative methods for solving the generalized Sylvester matrix equation. Dehghan and Hajarian [9] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (AY − ZB, CY − ZD) = (E, F) over reflexive matrices.
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