On a transformation of the ∗-congruence Sylvester equation for the least squares optimization. (2nd September 2020)

• Record Type: Journal Article
• Authors: Satake, Yuki; Sogabe, Tomohiro; Kemmochi, Tomoya; Zhang, Shao-Liang
• Is Part Of: Optimization methods and software. Volume 35:Number 5(2020)
• Page Start: 974
• Page End: 981
• Publication Date: 2020-09-02
• Subjects: The ★-congruence Sylvester equation -- the T-congruence Sylvester equation -- the ∗-congruence Sylvester equation; 15A24; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1734004
• Languages: English
• ISSNs: 1055-6788

Abstract

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.