On the reducibility of centrosymmetric matices - applications in engineering problems
This work concerns the class \(K^{N\times N}\) of centrosymmetric complex \(N\times N\) matrices and offers a number of results leading to significant computational saving in many engineering applications. A matrix \(R\in C^{N\times N}\) is said to be centrosymmetric if \(R=E_ NRE_ N\), where \(E_ N\) is the contra-identity matrix having ones along the crossdiagonal and zeros elsewhere. The class \(K^{N\times N}\) is closed under addition, multiplication, transposition, inversion and left multiplication by \(E_ N\) (Lemma 3). Every matrix \(R\in K^{N\times N}\) is similar to a quasi-diagonal matrixa \(\hat R=diag(F,G)\), where \(F\in C^{M\times M}\) and \(G\in C^{M\times M}\) for \(N=2M\), or \(G\in C^{(M+1)\times (M+1)}\) for \(N=2M+1\) (Lemma 4). The last Lemma reduces the problem of computing det(R) to two problems of computing det(F) and det(G) (Th. 1). For \(R\in K^{N\times N}\) having distinct eigenvalues from M linearly independent (l.i.) eigenvectors of F, M. l.i. skew-symmetric eigenvectors of R and from M, or \(M+1\) l.i. eigenvectors of G, M or \(M+1\) symmetric eigenvectors of R can be determined, where \(N=2M\), or \(N=2M+1\), respectively (Th. 2.). Thereby, a vector \(x\in C^{N\times 1}\) is said to be symmetric, or skew-symmetric if \(x=E_ Nx\), or \(x=-E_ Nx\), respectively. In general, for every eigenvalues \(\lambda\) of \(R\in K^{N\times N}\) there is a corresponding symmetric or skew-symmetric eigenvector x of R (see pp. 80-81). The converse of Th. 2 is also true: A matrix \(R\in C^{N\times N}\) having N l.i. eigenvectors which are either symmetric or skew-symmetric, is a centrosymmetric matrix (Th. 7). Some applications of the above results are given.
reducibility of centro-symmetric matrices
scientific article; zbMATH DE number 4103262
scientific article