Singular value decomposition for comb filter matrices Jesús Gutiérrez-Gutiérrez CEIT and Tecnun (University of Navarra), Manuel Lardizábal 15, 20018 San Sebastián, Spain article info Keywords: Eigenvalues Eigenvectors Pentadiagonal matrices Singular values Tridiagonal matrices abstract In this paper, we present an eigenvalue decomposition for any n n complex matrix with constant diagonals T n ¼ða jk Þ n j;k¼1 satisfying that there exists a positive integer m such that a k ¼ 0 for all k R fm; 0; mg. Moreover, from this eigenvalue decomposition we obtain a singular value decomposition for any comb filter matrix. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction In [1,2] Rimas presented an eigenvalue decomposition for the n n pentadiagonal Toeplitz matrix pentadiag n ð1; 0; 0; 0; 1Þ, where pentadiag n ða 2 ; a 1 ; a 0 ; a 1 ; a 2 Þ :¼ a 0 a 1 a 2 0 0  0 a 1 a 0 a 1 a 2 0  0 a 2 a 1 a 0 a 1 a 2  0 0 a 2 a 1 a 0 a 1  0 0 0 a 2 a 1 a 0 . . . . . . . . . . . . . . . . . . 0 0 0 0  a 0 0 B B B B B B B B B B @ 1 C C C C C C C C C C A ; with a 2 ; a 1 ; a 0 ; a 1 ; a 2 2 C, and where C denotes the set of (finite) complex numbers. In Section 2 of the present paper, we generalize that Rimas’ work to a much wider class of banded Toeplitz matrices, namely, we obtain an eigenvalue decompo- sition for any n n complex Toeplitz matrix T n ¼ða jk Þ n j;k¼1 satisfying that there exists a positive integer m such that a k ¼ 0 for all k R fm; 0; mg: T n ¼ a 0 0 0  0 a m 0  0 0 a 0 0  0 0 a m  0 0 0 a 0  0 0 0  0 . . . . . . . . . . . . . . . . . . . . . 0 0 0  a 0 0 0 a m 0 0  0 a 0 0 0 a m 0  0 0 a 0 . . . . . . . . . . . . . . . 0 0 0  a 0 0 B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C A : ð1Þ 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.050 E-mail address: jgutierrez@ceit.es Applied Mathematics and Computation 222 (2013) 472–477 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc