A note on computing matrix–vector products with generalized centrosymmetric (centrohermitian) matrices

Abstract

To reduce the costs of computing matrix–vector product Ax related to a centrosymmetric matrix A as compared to the case of an arbitrary matrix A, two algorithms were proposed recently, one was designed by Melman [A. Melman, Symmetric centrosymmetric matrix–vector multiplication, Linear Algebra Appl. 320 (2000) 193–198] for symmetric centrosymmetric matrices, another was presented by Fassbender and Ikramov [H. Fassbender, K.D. Ikramov, Computing matrix–vector products with centrosymmetric and centrohermitian matrices, Linear Algebra Appl. 364 (2003) 235–241] for general centrosymmetric matrices. In this note we further discuss this topic of computing Ax, where A is a generalized centrosymmetric matrix. We firstly investigate the reducibility of a generalized centrosymmetric matrix and then provide an algorithm which can be viewed as a generalization of one [H. Fassbender, K.D. Ikramov, Computing matrix–vector products with centrosymmetric and centrohermitian matrices, Linear Algebra Appl. 364 (2003) 235–241]. We show that our algorithm is suitable for computing many matrix–vector products with the same matrix. Furthermore, we show that similar results can be obtained for certain subclasses of generalized centrosymmetric matrices, as compared to the case of a centrosymmetric matrix, and analogue gains are available for generalized skew-centrosymmetric or generalized centrohermitian matrices.