Determinants of some pentadiagonal matrices

TL;DR

This paper derives explicit formulas for the determinants of specialized pentadiagonal matrices with structured sub- and superdiagonals, including Toeplitz and imperfect variants, and explores their factorization methods.

Contribution

It provides new explicit determinant formulas for a class of pentadiagonal matrices with structured diagonals and introduces factorization techniques for these matrices.

Findings

Explicit determinant formulas for structured pentadiagonal matrices.

Factorization methods for these matrices using Egerváry and Szász's rearrangement.

Analysis of Toeplitz and imperfect Toeplitz matrix variants.

Abstract

In this paper we consider pentadiagonal $(n+1)\times(n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and also consider the Toeplitz and "imperfect" Toeplitz versions of such matrices. Imperfectness means that the first and last $k$ elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerv\'ary and Sz\'asz we also show how these determinants can be factorized.