### 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 |

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### 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

*Linear algebra and its applications*,

*513*, 276-303. https://doi.org/10.1016/j.laa.2016.10.023

**Wreath product of matrices.** / D'Angeli, Daniele; Donno, Alfredo.

Research output: Contribution to journal › Article › Research › peer-review

*Linear algebra and its applications*, vol. 513, pp. 276-303. https://doi.org/10.1016/j.laa.2016.10.023

}

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 -