Factorizations of Matrices With Recursive Entries and Related Topics

TL;DR

This paper explores matrices defined by recursive relations, providing a general decomposition and linking them to Pascal-like and Toeplitz matrices, thus unifying various known matrix factorizations.

Contribution

It introduces a general decomposition framework for matrices with recursive entries, encompassing many known factorizations as special cases.

Findings

Derived a general matrix decomposition for recursive matrices

Connected recursive matrices to Pascal-like and Toeplitz matrices

Unified various known matrix factorizations under a common framework

Abstract

This article examines matrices whose entries are determined by recursive relations of the form $A_{i, j} = x A_{i, j-1} + y A_{i-1, j-1} + z A_{i-1, j}$, where $x, y, z$ are constants, and the initial conditions are defined along the first row and column. We present a general decomposition for such matrices and show that many of the known decompositions are particular cases of this more general decomposition. Additionally, we provide a decomposition of these matrices into Pascal-like matrices and a basic Toeplitz matrix.