The least-squares solutions of inconsistent matrix equation over symmetric and antipersymmetric matrices
Let \(SA^n\) denote the set of jointly symmetric and antipersymmetric matrices \(A\in{\mathbb R}^{n\times n}\) whose elements, by definition, satisfy the pair of formulae \(a_{ij}=a_{ji}\) and \(a_{ij}=-a_{n-j+1,n-i+1}\) for \(i,j=1,2,\ldots,n\). The authors use singular value decomposition to find explicitly the set \({\mathcal S}\subset SA^n\) of general solutions \(L\) which minimize \(\| LG-E\| \) (Problem 1) in the Frobenius norm for \(G,E\in{\mathbb R}^{n\times n}\). Next, given \(L^*\in{\mathbb R}^{n\times n}\), the unique solution \(\widehat L\in {\mathcal S}\) which minimizes \(\| L^*-\widehat L\| \) (Problem 2) in the Frobenius norm is also obtained by means of the singular value decomposition. In the course of the proofs, expressions for matrices in \(SA^{2k}\) and \(SA^{2k+1}\) are obtained in terms of explicit orthogonal similarities. Algorithms for matrices of even and odd orders are applied to numerical examples to illustrate the theory. In particular, as \(L^*\) approaches a solution to Problem 1, the norm \(\| L^*-\widehat L\| \) becomes smaller. References 8 to 19 are not quoted in the paper.
scientific article; zbMATH DE number 2073769
scientific article