Weyr structures of matrices and relevance to 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.