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Tensor product of GLT sequences

Carlo Garoni

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 $κ_1,\ldots,κ_d$, then their tensor (Kronecker) product $\{A_{n,1}\otimes\cdots\otimes A_{n,d}\}_n$ is a GLT sequence with symbol $κ_1\otimes\cdots\otimesκ_d$, 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.

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 are GLT sequences with symbols , then their tensor (Kronecker) product is a GLT sequence with symbol , up to suitable permutation matrices that only depend on the dimensions of the involved matrices . 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.
Paper Structure (20 sections, 15 theorems, 91 equations)

This paper contains 20 sections, 15 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Overview of the theory of GLT sequences
  3. Multi-index notation
  4. Singular value and spectral distribution of a sequence of matrices
  5. Special sequences of matrices
  6. Zero-distributed sequences
  7. Sequences of diagonal sampling matrices
  8. Toeplitz sequences
  9. Approximating classes of sequences
  10. GLT sequences
  11. General properties of tensor products
  12. Tensor product of matrices
  13. Tensor product of functions
  14. S.u. sequences of matrices and tensor products
  15. Tensor product of a.c.s.
  16. ...and 5 more sections

Key Result

Lemma 3.1

For all matrices $X_1\in\mathbb C^{m_1\times n_1}$ and $X_2\in\mathbb C^{m_2\times n_2}$, we have

Theorems & Definitions (37)

  • Definition 2.1: singular value and spectral distribution of a sequence of matrices
  • Definition 2.2: approximating class of sequences
  • Definition 2.3: generalized locally Toeplitz sequence
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 27 more