On Vertically-Recurrent Matrices and Their Algebraic Properties

TL;DR

This paper introduces a new class of vertically-recurrent matrices, explores their algebraic properties including decomposition and powers, and demonstrates their applications in matrix decomposition and ladder network theory.

Contribution

It defines vertically-recurrent matrices, derives their lower triangular decomposition, analyzes their powers, and applies them to matrix decomposition and ladder network problems.

Findings

Derived a formula for lower triangular decomposition.

Analyzed powers of vertically-recurrent matrices.

Applied these matrices to matrix decomposition and ladder networks.

Abstract

In this paper, we first introduce the new class of vertically-recurrent matrices, using a generalization of "the Hockey stick and Puck theorem" in Pascal's triangle. Then, we give an interesting formula for the lower triangular decomposition of these matrices. We also deal with the $m$-th power of these matrices in some special cases. Furthermore, we present two important applications of these matrices for decomposing \emph{admissible matrices} and matrices which arise in the theory of \emph{ladder networks}. Finall,y we pose some open problems and conjectures about these new kind of matrices.