On a transformation of the ∗-congruence Sylvester equation for the least squares optimization. (2nd September 2020) - Satake, Yuki; Sogabe, Tomohiro; Kemmochi, Tomoya; Zhang, Shao-Liang
Abstract : The ★ -congruence Sylvester equation is the matrix equation A X + X ⋆ B = C, where A ∈ F m × n, B ∈ F n × m and C ∈ F m × m are given, whereas X ∈ F n × m is to be determined. Here, F = R or C, and ⋆ = T (transposed) or ∗ (conjugate transposed). Very recently, Satake et al. showed that under some conditions, the matrix equation for the case ⋆ = T is equivalent to the generalized Sylvester equation. In this paper, we demonstrate that the result can be extended to the case ⋆ = ∗ . Through this extension, the least squares solution of the ∗ -congruence Sylvester equation may be obtained using well-researched results on the least squares solution of the generalized Sylvester equation.