Remarks on band matrices

Abstract

In this note we consider band- or tridiagonal-matrices of orderk whose elements above, on, and below the diagonal are denoted byb _i ,a _i,c _i . In the periodic case, i.e.b _i+m =b _i etc., we derive fork=nm andk=nm−1 formulas for the characteristic polynomial and the eigenvectors under the assumption that$\mathop \prod \limits_{i = 1}^m c_i b_i > 0$ i=1 In the latter case it is shown that the characteristic polynomial is divisible by them−1-th minor, as was already observed byRósa. We also give estimations for the number of real roots and an application to Fibonacci numbers.