Elementary triangular matrices and inverses of $k$-Hessenberg and triangular matrices

TL;DR

This paper explores properties of triangular matrices using elementary matrices, providing explicit formulas for inverses of strict $k$-Hessenberg and banded matrices, with extensions to block matrices.

Contribution

It introduces new factorization and inversion formulas for strict $k$-Hessenberg and banded matrices using elementary triangular matrices.

Findings

Explicit inverse formulas for strict $k$-Hessenberg matrices

Factorization and multiplication properties of triangular matrices

Extensions to block triangular and block Hessenberg matrices

Abstract

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict $k$-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices.