2015/07/09 by Daniele D'Angeli, Daniele D’Angeli, Alfredo Donno · 9 citations
Computer Science · Mathematics · #Adjacency list #Adjacency matrix #Algebra over a field #Circulant matrix #Combinatorics #Discrete mathematics #Geometry #Graph #Graph theory and applications #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Matrix multiplication #Product (mathematics) #Pure mathematics #Quantum #Random Matrices and Applications #Sylvester matrix #Uniqueness #Wreath product #math.RA #msc:05C50 #msc:05C76 #msc:05C81 #msc:15A18 #msc:15A69
paper · pdf · doi:10.1016/j.laa.2016.10.023
published in Linear Algebra and its Applications 513, 276-303 (Elsevier BV) · 25 pages
arxiv created 2015/07/09 · openalex publication_date 2016/10/27 · arxiv updated 2016/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 generalized Sylvester matrix equations is treated.