# Weyr structures of matrices and relevance to commutative finite-dimensional algebras

## TL;DR

This paper explores the Weyr structure of matrices, especially the Sierpinski matrices, and connects these findings to properties of commutative finite-dimensional algebras, providing a simpler derivation of known structures.

## Contribution

It introduces a new method to determine the Weyr structure of certain block matrices, simplifying previous complex proofs and linking matrix theory to algebraic structures.

## Key findings

- Weyr structure of Sierpinski matrices given by binomial coefficients
- Simplified derivation of matrix structures compared to algebraic geometry methods
- Implications for understanding commutative finite-dimensional algebras

## Abstract

We relate the Weyr structure of a square matrix $B$ to that of the $t \times t$ block upper triangular matrix $C$ that has $B$ down the main diagonal and first superdiagonal, and zeros elsewhere. Of special interest is the case $t = 2$ and where $C$ is the $n$th Sierpinski matrix $B_n$, which is defined inductively by $B_0 = 1$ and $B_n = \left[\begin{array}{cc} B_{n-1} & B_{n-1} 0 & B_{n-1} \end{array} \right]$. This yields an easy derivation of the Weyr structure of $B_n$ as the binomial coefficients arranged in decreasing order. Earlier proofs of the Jordan analogue of this had often relied on deep theorems from such areas as algebraic geometry. The result has interesting consequences for commutative, finite-dimension algebras.