Explicit determinantal formula for a class of banded matrices

Abstract

In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

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…”

On determinantal recurrence relations of banded matrices

“…Both conjectures have recently attracted the attention of many researchers (Amanbek et al, 2020;Kurmanbek et al, 2020;Shitov, 2021).…”

Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

“…The result is summarized in Theorem 9 below: Theorem 9. Let A n be given as in (1), with a 0 = 56 and a 1 = −39. Then the following inequality holds for p ∈ {1, 2, ∞}:…”

“…In addition to the algorithmic development, many authors have contributed to the inverse properties of banded Toeplitz and near Toeplitz matrices, such as exact inverse formulas [10,5,22,17,23], bounds for entries of the inverse matrices, and bounds for the inverse norm. Examining formulas for determinant of such matrices can be also useful to explore the existence and uniqueness of solution related to the ordinary or partial differential problems [1,4,3,12,15].…”