The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras

TL;DR

This paper explores the properties of the Pascal matrix, identifying commuting tridiagonal matrices, analyzing their algebraic relations, and demonstrating a stable diagonalization method using eigenvectors.

Contribution

It explicitly constructs symmetric tridiagonal matrices commuting with the Pascal matrix and analyzes their algebraic relations and eigenstructure.

Findings

Explicit expressions for commuting tridiagonal matrices are provided.

All linear relations of Pascal matrix entries derive from three basic relations.

A numerically stable diagonalization method using eigenvectors is demonstrated.

Abstract

We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.