The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation

• Published in: Journal of computational and applied mathematics 2006-12, Vol.197 (1), p.44-52
• Main Authors: Gong, Lisha, Hu, Xiyan, Zhang, Lei
• Format: Article
• Language: English
• Subjects: Algebra; Anti-symmetric matrix; Exact sciences and technology; Frobenius norm; Linear and multilinear algebra, matrix theory; Linear constraint; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Optimal approximation; Sciences and techniques of general use
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2005.10.021

This paper mainly discusses the following two problems: Problem I Given A ∈ R n × m , B ∈ R m × m , X 0 ∈ ASR q × q (the set of q × q anti-symmetric matrices), find X ∈ ASR n × n such that A T XA = B , X 0 = X ( [ 1 : q ] ) , where X ( [ 1 : q ] ) is the q × q leading principal submatrix of matrix X. Problem II Given X * ∈ R n × n , find X ^ ∈ S E such that ∥ X * - X ^ ∥ = min X ∈ S E ∥ X * - X ∥ , where ∥ · ∥ is the Frobenius norm, and S E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. Moreover, the optimal approximation solution, an algorithm and a numerical example of Problem II are provided.