A Fiedler's type characterization of band matrices

Abstract

Let K be a field and p an integer positive number. We denote by Bpn (K) the set of n-by-n symmetric band matrices of bandwidth 2p − 1, i.e., if A = [aij ] ∈ Bpn (K) then aij = 0 if |i − j| > p − 1. Let b Bpn (K) be the set of matrices from Bpn (K) in which the entries (i, j), |i − j| = p − 1, are different from zero. Let A be a n-by-n symmetric matrix with entries from K; and p such that 3 6 p 6 n. We will show that: rank(A + B) > n − p + 1, for every B ∈ Bp−1 n (K), if and only if A ∈ b Bpn (K).