Tensor product of GLT sequences

TL;DR

This paper proves that the tensor product of GLT sequences results in a new GLT sequence with a symbol given by the tensor product of the original symbols, including explicit permutation matrices for the transformation.

Contribution

It establishes that the tensor product of GLT sequences is itself a GLT sequence with a computable symbol, extending the algebraic properties of GLT sequences.

Findings

Tensor product of GLT sequences yields a GLT sequence.

The symbol of the tensor product is the tensor product of individual symbols.

Permutation matrices for the tensor product are explicitly constructed.

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.