Fast block diagonalization of k-tridiagonal matrices
abstract In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonalmatrices. The block diagonalization provides us with some useful results: e.g., another der-ivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy,T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456–4461]; efﬁcient (symbolic) algorithm for computing the matrix determinant. 2011 Elsevier Inc. All rights reserved. 1. Introduction and objectivesLet T ðkÞn be the k-tridiagonal matrix of order n n, i.e.T ðkÞn :¼d 1 0 0 a 1 0 00 d 2 0 ...0 a 2 ... ...... 0 ...0 ... ... ...00 ... ...d n k ... ... ...a n k b kþ1 0... ... ... ... ...00 b kþ2 ... ...0 ...0 ...... ... ...0 ...0 d n 1 00 0 b n 0 0 d n 0BBBBBBBBBBBBBBBBBBB@1CCCCCCCCCCCCCCCCCCCA: ð1ÞFor example, T ð3Þ7 is given as follows:T ð3Þ7 ¼d 1 00a 1 00 00 d 2 00a 2 0000d 3 00a 3 0b 4 00d 4 00a 4 0 b 5 00d 5 0000b 6 00d 6 0000b 7 00d 7 0BBBBBBBBBBB@1CCCCCCCCCCCA: ð2Þ 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2011.08.014