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GLT matrix-sequences and few emblematic applications

Muhammad Faisal Khan

TL;DR

This thesis develops and applies Generalized Locally Toeplitz (GLT) theory to the spectral analysis of matrix-sequences, focusing on the geometric mean of Hermitian positive definite GLT sequences. It proves that, for commuting symbols, the geometric mean sequence is GLT with symbol equal to the geometric mean of the input symbols (and extends this to multilevel and block settings, including non-invertible cases under suitable conditions); it also proposes a Karcher-mean generalization for more than two matrices. The work provides extensive numerical evidence in 1D/2D settings, including Galerkin discretizations and spline-based discretizations, illustrating how the spectral distributions align with GLT symbols and exploring extremal eigenvalue behavior. A central application is to mean-field quantum spin systems, notably the Curie–Weiss model, where CW Hamiltonians form GLT sequences with explicitly computable spectral distributions, linking abstract GLT theory to concrete physical models. The results establish maximality of the commuting/invertibility assumptions and introduce momentary GLT symbols to capture finite-size spectral features, outlining open problems for noncommuting or degenerate cases and for extending Karcher-mean results to more than two sequences. Overall, the work solidifies GLT as a versatile tool for understanding the asymptotic spectral behavior of complex structured matrices arising in PDE discretizations and quantum models, with tangible implications for preconditioning, discretization design, and spectral phenomenology in physics.

Abstract

This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) $*$-algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT sequences and its applications in mathematical physics. For two HPD sequences $\{A_n\}_n \sim_{\mathrm{GLT}} κ$ and $\{B_n\}_n \sim_{\mathrm{GLT}} ξ$ in the same $d$-level, $r$-block GLT $*$-algebra, we prove that when $κ$ and $ξ$ commute, the geometric mean sequence $\{G(A_n,B_n)\}_n$ is GLT with symbol $(κξ)^{1/2}$, without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for $r=1$, $d\ge1$. In degenerate cases, we identify conditions ensuring $\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} G(κ,ξ)$. For $r>1$ and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to $k\ge2$ sequences via the Karcher mean, obtaining $\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(κ_1,\ldots,κ_k)$. Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions.

GLT matrix-sequences and few emblematic applications

TL;DR

This thesis develops and applies Generalized Locally Toeplitz (GLT) theory to the spectral analysis of matrix-sequences, focusing on the geometric mean of Hermitian positive definite GLT sequences. It proves that, for commuting symbols, the geometric mean sequence is GLT with symbol equal to the geometric mean of the input symbols (and extends this to multilevel and block settings, including non-invertible cases under suitable conditions); it also proposes a Karcher-mean generalization for more than two matrices. The work provides extensive numerical evidence in 1D/2D settings, including Galerkin discretizations and spline-based discretizations, illustrating how the spectral distributions align with GLT symbols and exploring extremal eigenvalue behavior. A central application is to mean-field quantum spin systems, notably the Curie–Weiss model, where CW Hamiltonians form GLT sequences with explicitly computable spectral distributions, linking abstract GLT theory to concrete physical models. The results establish maximality of the commuting/invertibility assumptions and introduce momentary GLT symbols to capture finite-size spectral features, outlining open problems for noncommuting or degenerate cases and for extending Karcher-mean results to more than two sequences. Overall, the work solidifies GLT as a versatile tool for understanding the asymptotic spectral behavior of complex structured matrices arising in PDE discretizations and quantum models, with tangible implications for preconditioning, discretization design, and spectral phenomenology in physics.

Abstract

This thesis advances the spectral theory of structured matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) -algebras, focusing on the geometric mean of Hermitian positive definite (HPD) GLT sequences and its applications in mathematical physics. For two HPD sequences and in the same -level, -block GLT -algebra, we prove that when and commute, the geometric mean sequence is GLT with symbol , without requiring invertibility of either symbol, settling \cite[Conjecture 10.1]{garoni2017} for , . In degenerate cases, we identify conditions ensuring . For and non-commuting symbols, numerical evidence shows the sequence still admits a spectral symbol, indicating maximality of the commuting result. Numerical experiments in scalar and block settings confirm the theory and illustrate spectral behaviour. We also sketch the extension to sequences via the Karcher mean, obtaining . Finally, we apply the GLT framework to mean-field quantum spin systems, showing that matrices from the quantum Curie--Weiss model form GLT sequences with explicitly computable spectral distributions.

Paper Structure

This paper contains 77 sections, 24 theorems, 254 equations, 40 figures, 16 tables.

Table of Contents

  1. Matrices and matrix-sequences
  2. Singular value and eigenvalue distributions of a matrix-sequence
  3. Approximating classes of sequences
  4. Multilevel discretization matrices
  5. Structured matrices
  6. Toeplitz matrices
  7. Relationship between the Toeplitz matrix and its generating function
  8. Circulant matrices
  9. Diagonalization and computational efficiency of circulant matrices
  10. omega-circulant matrices
  11. Block omega-circulant matrices
  12. tau matrices
  13. Spectral properties and the discrete sine transform
  14. Hankel matrices
  15. GLT structures in matrix-sequences
  16. ...and 62 more sections

Key Result

Proposition 1.2

Let $\{A_{n}\}_n$ be a matrix-sequence such that $A_{n} = B_{n} + C_{n}$, where $B_{n}$ is Hermitian $\forall n \in \mathbb{N}$. Suppose that Then $\{A_{n}\}_n \sim_\lambda \psi$.

Figures (40)

  • Figure 9: 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)
  • Figure 10: Case 1 - Example 2. Comparison of the ordered eigenvalues of $G(A_n,B_n)$ (colored markers) with the rearranged values of $\widetilde{G}(\kappa,\xi)$ (solid red line) for $n=40,80,160,320$.
  • Figure 11: Case 2, Example 1. Sorted eigenvalues of $G(A_n,B_n)$ (colored markers) versus the rearranged eigenvalue distribution predicted by $G(\kappa,\xi)$ (solid red line), for $n=40,80,160,320$.
  • Figure 12: Case 2, Example 2. Sorted eigenvalues of $G(A_n,B_n)$ (colored markers) versus the rearranged eigenvalue distribution from $G(\kappa,\xi)$ (solid red line), for $n=40,80,160,320$.
  • Figure :
  • ...and 35 more figures

Theorems & Definitions (42)

  • Definition 1
  • Remark 1.1
  • Proposition 1.2: Barbarino2020
  • Definition 2: Approximating class of sequences
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 32 more