A periodic determinantal property for (0,1) double banded matrices

Abstract

In this paper, we extend the notion of banded matrices to matrices where two bands are allowed, called double banded matrices. Our main aim is to establish the periodicity of the determinants for (0, 1) double banded matrices. As a corollary, we answer to two recent conjectures and other extensions. Several illustrative examples are provided as well.

The determinants of certain double banded (0,1) Toeplitz matrices

“…It is worth indicating that here we need the precondition that n c + 6, or equivalently, n max{c + 6, 6 − c}, which guarantees the using of Theorem 1 to the three determinants ∆ 1 ( * , * ). Then the expression of ∆ 2 n, n+c As to the remaining two cases n = c + 2 and n = c + 4, they just correspond to the two conjectures proposed in [2], which were confirmed in [6] (see also [1,4,5,7]. Following the notations in the paper, they claim that…”

“…Set ∆ s,t (n, r) = det A s,t (n, r). Now we recall several formulae obtained in our previous publication [4,Theorems 3.3 and 3.4], which lead to an explicit expression of ∆ 1,t (n, r), in terms of the determinants of form ∆ 1 ( * ).…”