The Hermitian R-symmetric Solutions of the Matrix Equation AXA∗ = B

In this paper, we mainly discuss solving the following problems. Problem I. Given matrices A ∈ Cm×n and B ∈ Cm×m, find X ∈ HRSn×n such that AXA∗ = B, where HRSn×n = {X ∈ Hn×n|RXR = X, for given R ∈ Cn×n satisfying R = R∗ = R−1 = In}. Problem II. Given a matrix X̃ ∈ Cn×n, find X̂ ∈ SE such that ‖X̃ − X̂‖F = inf X∈SE ‖X̃ − X‖F , where ‖ · ‖ is the Frobenius norm, and SE is the solution set of Problem I. Expressions for the general solution of Problem I are derived. Necessary and sufficient conditions for the solvability of Problem I are provided. For Problem II, an expression for the solution is given as well. Mathematics Subject Classification: 65F15, 65F20