TY - JOUR

T1 - Decompositions and eigenvectors of Riordan matrices

AU - Cheon, Gi Sang

AU - Cohen, Marshall M.

AU - Pantelidis, Nikolaos

N1 - Funding Information:
The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2019R1A2C1007518, 2016R1A5A1008055).
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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×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 three different types of eigenvectors. Finally, given a nonzero vector v, we investigate the Riordan matrices A that stabilize the vector v, i.e. Av=v.

AB - 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×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 three different types of eigenvectors. Finally, given a nonzero vector v, we investigate the Riordan matrices A that stabilize the vector v, i.e. Av=v.

KW - Eigenvectors of Riordan matrix

KW - Formal series of infinite order

KW - Stabilizers

UR - http://www.scopus.com/inward/record.url?scp=85125297443&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2022.02.023

DO - 10.1016/j.laa.2022.02.023

M3 - Article

AN - SCOPUS:85125297443

SN - 0024-3795

VL - 642

SP - 118

EP - 138

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -