Wreath product of matrices

Daniele D'Angeli, Alfredo Donno*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

We introduce a new matrix product, that we call the wreath product of matrices. The name is inspired by the analogous product for graphs, and the following important correspondence is proven: the wreath product of the adjacency matrices of two graphs provides the adjacency matrix of the wreath product of the graphs. This correspondence is exploited in order to study the spectral properties of the famous Lamplighter random walk: the spectrum is explicitly determined for the “Walk or switch” model on a complete graph of any size, with two lamp colors. The investigation of the spectrum of the matrix wreath product is actually developed for the more general case where the second factor is a circulant matrix. Finally, an application to the study of the uniqueness of the solution of generalized Sylvester matrix equations is treated.

Original languageEnglish
Pages (from-to)276-303
Number of pages28
JournalLinear algebra and its applications
Volume513
DOIs
Publication statusPublished - 15 Jan 2017

Keywords

  • Block-circulant matrix
  • Circulant matrix
  • Lamplighter random walk
  • Wreath product of graphs
  • Wreath product of matrices

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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