Tensor product of GLT sequences

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. Sequences of this kind, which are known as GLT sequences, arise in several applications, including the discretization of differential and integral equations. Associated with any GLT sequence is a special function called symbol. In this paper, we prove that, if $\{A_{n,1}\}_n,\ldots,\{A_{n,d}\}_n$ are GLT sequences with symbols $\kappa_1,\ldots,\kappa_d$, then their tensor (Kronecker) product $\{A_{n,1}\otimes\cdots\otimes A_{n,d}\}_n$ is a GLT sequence with symbol $\kappa_1\otimes\cdots\otimes\kappa_d$, up to suitable permutation matrices that only depend on the dimensions of the involved matrices $A_{n,1},\ldots,A_{n,d}$. The permutation matrices in question are explicitly defined through a recursive formula that allows for their algorithmic computation. Some applications of the presented result are discussed.