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 language | English |
|---|---|
| Pages (from-to) | 276-303 |
| Number of pages | 28 |
| Journal | Linear algebra and its applications |
| Volume | 513 |
| DOIs | |
| Publication status | Published - 15 Jan 2017 |
Fingerprint
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
Cite this
Wreath product of matrices. / D'Angeli, Daniele; Donno, Alfredo.
In: Linear algebra and its applications, Vol. 513, 15.01.2017, p. 276-303.Research output: Contribution to journal › Article › Research › peer-review
}
TY - JOUR
T1 - Wreath product of matrices
AU - D'Angeli, Daniele
AU - Donno, Alfredo
PY - 2017/1/15
Y1 - 2017/1/15
N2 - 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.
AB - 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.
KW - Block-circulant matrix
KW - Circulant matrix
KW - Lamplighter random walk
KW - Wreath product of graphs
KW - Wreath product of matrices
UR - http://www.scopus.com/inward/record.url?scp=84994236336&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2016.10.023
DO - 10.1016/j.laa.2016.10.023
M3 - Article
VL - 513
SP - 276
EP - 303
JO - Linear algebra and its applications
JF - Linear algebra and its applications
SN - 0024-3795
ER -