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Geometric means of HPD GLT matrix-sequences: a maximal result beyond invertibility assumptions on the GLT symbols

Asiim Ilyas, Muhammad Faisal Khan, Valerio Loi, Stefano Serra-Capizzano

TL;DR

This work analyzes the spectral distribution of geometric mean matrix-sequences G(A_n,B_n) formed from HPD GLT sequences with GLT symbols κ and ξ. It proves that when the GLT symbols commute, the geometric mean remains within the GLT class and satisfies {G(A_n,B_n)} ∼GLT (κ ξ)^{1/2} without requiring invertibility of κ or ξ, extending prior results to the d-level, r-block GLT setting. In general noncommuting or rank-deficient cases, the invertibility assumption may be essential, and numerical experiments reveal maximal behavior and the emergence of momentary symbols that can alter the asymptotic distribution. The paper also introduces the candidate symbol tilde G(κ,ξ) capturing the intersection of the essential ranges and discusses conjectured GLT closure under the geometric mean, supported by extensive simulations across commuting and noncommuting scenarios. Overall, the results advance GLT theory by clarifying when matrix means preserve asymptotic structure and by highlighting rich finite-size spectral phenomena via momentary symbols.

Abstract

In the current work, we consider the study of the spectral distribution of the geometric mean matrix-sequence of two matrix-sequences $\{G(A_n, B_n)\}_n$ formed by Hermitian Positive Definite (HPD) matrices, assuming that the two input matrix-sequences $\{A_n\}_n, \{B_n\}_n$ belong to the same $d$-level $r$-block Generalized Locally Toeplitz (GLT) $\ast$-algebra with $d,r\ge 1$ and with GLT symbols $κ, ξ$. Building on recent results in the literature, we examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere (a.e.) is necessary. Since inversion is mainly required due to the non-commutativity of the matrix product, it was conjectured that the hypothesis on the invertibility of the GLT symbols can be removed. In fact, we prove the conjectured statement that is \[ \{G(A_n, B_n)\}_n \sim_{\mathrm{GLT}} (κξ)^{1/2} \] when the symbols $κ, ξ$ commute, which implies the important case where $r=1$ and $d \geq 1 $, while the statement is generally false or even not well posed when the symbols are not invertible a.e. and do not commute. In fact, numerical experiments are conducted in the case where the two symbols do not commute, showing that the main results of the present work are maximal. Further numerical experiments, visualizations, and conclusions end the present contribution.

Geometric means of HPD GLT matrix-sequences: a maximal result beyond invertibility assumptions on the GLT symbols

TL;DR

This work analyzes the spectral distribution of geometric mean matrix-sequences G(A_n,B_n) formed from HPD GLT sequences with GLT symbols κ and ξ. It proves that when the GLT symbols commute, the geometric mean remains within the GLT class and satisfies {G(A_n,B_n)} ∼GLT (κ ξ)^{1/2} without requiring invertibility of κ or ξ, extending prior results to the d-level, r-block GLT setting. In general noncommuting or rank-deficient cases, the invertibility assumption may be essential, and numerical experiments reveal maximal behavior and the emergence of momentary symbols that can alter the asymptotic distribution. The paper also introduces the candidate symbol tilde G(κ,ξ) capturing the intersection of the essential ranges and discusses conjectured GLT closure under the geometric mean, supported by extensive simulations across commuting and noncommuting scenarios. Overall, the results advance GLT theory by clarifying when matrix means preserve asymptotic structure and by highlighting rich finite-size spectral phenomena via momentary symbols.

Abstract

In the current work, we consider the study of the spectral distribution of the geometric mean matrix-sequence of two matrix-sequences formed by Hermitian Positive Definite (HPD) matrices, assuming that the two input matrix-sequences belong to the same -level -block Generalized Locally Toeplitz (GLT) -algebra with and with GLT symbols . Building on recent results in the literature, we examine whether the assumption that at least one of the input GLT symbols is invertible almost everywhere (a.e.) is necessary. Since inversion is mainly required due to the non-commutativity of the matrix product, it was conjectured that the hypothesis on the invertibility of the GLT symbols can be removed. In fact, we prove the conjectured statement that is when the symbols commute, which implies the important case where and , while the statement is generally false or even not well posed when the symbols are not invertible a.e. and do not commute. In fact, numerical experiments are conducted in the case where the two symbols do not commute, showing that the main results of the present work are maximal. Further numerical experiments, visualizations, and conclusions end the present contribution.
Paper Structure (41 sections, 6 theorems, 64 equations, 8 figures, 10 tables)

This paper contains 41 sections, 6 theorems, 64 equations, 8 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Spectral tools
  3. Notation and terminology
  4. Multi-index notation
  5. Singular Value and Eigenvalue Distributions of a Matrix-Sequence
  6. Approximating Classes of Sequences
  7. Matrix-Sequences with Explicit or Hidden (Asymptotic) Structure
  8. Zero-distributed sequences
  9. Multilevel block Toeplitz matrices
  10. Block diagonal sampling matrices
  11. The $*$-Algebra of $d$-Level $r$-block GLT Matrix-Sequences
  12. Geometric Mean of GLT matrix-sequences
  13. Numerical experiments
  14. Validation of the distribution results
  15. Example 1
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\{A_n\}_n, \{B_{n,j}\}_n$, with $j,n \in \mathbb{N}$, be matrix-sequences and let $\psi, \psi_j : D \subset \mathbb{R}^d \to \mathbb{C}$ be measurable functions defined on a set $D$ with positive and finite Lebesgue measure. Suppose that Then Moreover, if all the involved matrices are Hermitian, the first assumption is replaced by $\{B_{n,j}\}_n \sim_{\lambda} \psi_j \; \text{for every } j,

Figures (8)

  • Figure 1: (a) n=40
  • Figure 2: (b) n=80
  • Figure 3: (a) n=40
  • Figure 4: (b) n=80
  • Figure 5: Case 1 — Example 1. Sorted eigenvalues of $G(A_n,B_n)$ (colored markers; $n=40,80,160,320$) versus the rearranged distribution of the symbol $\widetilde{G}(\kappa,\xi)$ (solid red line)
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 2: Intuition on the generalization
  • Theorem 5
  • ...and 5 more