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arxiv: 2604.03823 · v1 · submitted 2026-04-04 · 🧮 math.NA · cs.NA

A note on the spectral distribution of non-Hermitian block matrices with Toeplitz blocks

Andrea Adriani , Giacomo Tento This is my paper

Pith reviewed 2026-05-13 16:51 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords spectral distributionnon-Hermitian matricesToeplitz blocksGLT sequencesgeometric meanblock matriceslimiting spectrum
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The pith

The limiting spectral distribution of non-Hermitian block matrices with Toeplitz blocks is given by the geometric mean of the associated GLT symbols.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the limiting spectral distribution for sequences of non-Hermitian matrices built from Toeplitz blocks. It combines the geometric mean of matrices with generalized locally Toeplitz sequence theory to derive the distribution in the non-symmetric setting. A reader would care because the result gives a way to predict eigenvalue locations for large structured matrices without computing the full spectrum. The derivations are supported by numerical tests that visualize the match between theory and computed eigenvalues. This extends earlier spectral results that required Hermitian symmetry.

Core claim

For non-Hermitian block matrix sequences whose blocks are Toeplitz, the empirical spectral distribution converges to the distribution induced by the geometric mean of the matrix-valued GLT symbol of the sequence.

What carries the argument

The geometric mean of matrices applied to the GLT symbol of the block matrix sequence.

Load-bearing premise

The block matrix sequences admit a GLT symbol whose geometric mean correctly describes the limiting spectral distribution for the non-Hermitian case.

What would settle it

A numerical experiment on a sequence of large non-Hermitian block Toeplitz matrices in which the empirical spectral measure deviates from the measure predicted by the geometric mean of the GLT symbol.

Figures

Figures reproduced from arXiv: 2604.03823 by Andrea Adriani, Giacomo Tento.

Figure 4.1
Figure 4.1. Figure 4.1: Comparison between the real part of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p014_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Comparison between the real part of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p014_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Comparison between the real part of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p015_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Comparison between the real part of the spectrum of [PITH_FULL_IMAGE:figures/full_fig_p015_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: In each column, depending on n, it is represented the value of the imaginary part of all the eigenvalues of An n maxλ |ℑ(λ(An))| 24 0.0113 48 0.0062 96 0.0035 192 0.0020 384 9.3026e-04 768 4.9599e-04 1536 2.4318e-04 [PITH_FULL_IMAGE:figures/full_fig_p016_4_5.png] view at source ↗
read the original abstract

In the present paper, we are concerned with the study of the spectral distribution of matrix-sequences showing a non-Hermitian block structure with Toeplitz blocks. We use the notion of geometric mean of matrices and the theory of Generalized Locally Toeplitz (GLT) sequences to perform our analysis and produce some numerical tests and visualizations to confirm our theoretical derivations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper studies the limiting spectral distribution of sequences of non-Hermitian block matrices with Toeplitz blocks. It combines the geometric mean of matrices with Generalized Locally Toeplitz (GLT) theory to derive an effective symbol for the block structure and supports the claim with numerical tests and visualizations.

Significance. If the geometric-mean construction can be rigorously justified for non-Hermitian symbols, the result would extend GLT techniques to a new class of block-structured non-Hermitian matrices, offering a compact description of their eigenvalue distributions with possible utility in numerical linear algebra.

major comments (1)
  1. The central construction relies on the geometric mean of the GLT symbols of the individual Toeplitz blocks to obtain the limiting eigenvalue distribution of the non-Hermitian block matrix sequence. The classical matrix geometric mean A#B is defined via the principal square root and requires both arguments to be positive definite Hermitian; no alternative definition (e.g., via holomorphic functional calculus on the joint spectrum or via the GLT symbol directly) is supplied for general complex-valued symbols. Because the blocks are arbitrary Toeplitz (hence their symbols need not be Hermitian or positive), the step that equates the geometric mean of the symbols to the effective GLT symbol of the block matrix is formally unsupported. This is load-bearing: without it, the claimed spectral distribution formula does not follow from standard GLT theory.
minor comments (1)
  1. The abstract states that numerical tests 'confirm our theoretical derivations' but provides no details on matrix dimensions, sampling of the symbol, or quantitative error measures used in the visualizations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for identifying the need to clarify the definition of the geometric mean in our non-Hermitian setting. We provide a point-by-point response below and will update the manuscript to include the missing definition.

read point-by-point responses

  1. Referee: The central construction relies on the geometric mean of the GLT symbols of the individual Toeplitz blocks to obtain the limiting eigenvalue distribution of the non-Hermitian block matrix sequence. The classical matrix geometric mean A#B is defined via the principal square root and requires both arguments to be positive definite Hermitian; no alternative definition (e.g., via holomorphic functional calculus on the joint spectrum or via the GLT symbol directly) is supplied for general complex-valued symbols. Because the blocks are arbitrary Toeplitz (hence their symbols need not be Hermitian or positive), the step that equates the geometric mean of the symbols to the effective GLT symbol of the block matrix is formally unsupported. This is load-bearing: without it, the claimed spectral distribution formula does not follow from standard GLT theory.

    Authors: We agree with the referee that the classical definition of the matrix geometric mean requires positive definite Hermitian matrices and that our manuscript does not explicitly provide an alternative definition for general complex symbols. This is indeed a gap in the current presentation. In the revised version of the paper, we will introduce a section on matrix functions where we define the geometric mean of two matrices A and B (with appropriate spectra) as exp( (log A + log B)/2 ), using the principal branch of the matrix logarithm via holomorphic functional calculus. This definition extends naturally to the pointwise application on the GLT symbols, which are continuous matrix-valued functions. We will also discuss the conditions under which this is well-defined for the symbols of Toeplitz blocks. With this addition, the application of GLT theory to derive the spectral distribution follows as stated. We thank the referee for this suggestion, which will improve the rigor of the work. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external GLT theory and matrix geometric mean without self-referential reduction

full rationale

The paper's abstract states that the analysis of the spectral distribution for non-Hermitian block matrices with Toeplitz blocks relies on the geometric mean of matrices combined with Generalized Locally Toeplitz (GLT) sequence theory, followed by numerical confirmation. No equations or steps are exhibited that reduce a claimed prediction or limiting distribution to a fitted parameter or self-defined quantity by construction. The central construction invokes prior GLT results as an external framework rather than deriving them internally, and the geometric mean is presented as a standard tool applied to the symbols without evidence of ansatz smuggling or uniqueness imported via self-citation chains. The derivation chain remains self-contained against external benchmarks, with numerical tests serving as independent verification rather than tautological confirmation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of GLT theory to the described block structure and on the geometric mean correctly capturing the non-Hermitian spectral symbol; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption GLT theory extends to non-Hermitian block matrices with Toeplitz blocks
    The analysis invokes GLT sequences to obtain the spectral distribution.

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