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

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.