Decompositions and eigenvectors of Riordan matrices

TL;DR

This paper explores the linear algebraic properties of Riordan matrices, including eigenvectors, eigenvalues, and stabilization, revealing new structural insights and classifications within this mathematical framework.

Contribution

It provides a detailed analysis of eigenvectors, eigenvalues, and the structure of Riordan matrices using formal power series interactions, including conditions for pseudo-involutions and vector stabilization.

Findings

Singular values of pseudo-involution Riordan matrices come in reciprocal pairs.

Riordan matrices can be classified into three types based on their eigenvector properties.

Conditions for the existence of eigenvectors and matrices stabilizing a given vector are characterized.

Abstract

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $n\times n$ pseudo-involution then the singular values of $A$ must come in reciprocal pairs. Moreover, we give a complete analysis of existence and nonexistence of the eigenvectors of Riordan matrices. This leads to a surprising partition of the group of Riordan matrices into matrices with three different types of eigenvectors. Finally, given a nonzero vector $v$, we investigate the Riordan matrices $A$ that stabilize the vector $v$, t i.e. $Av=v$.