  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.
