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arxiv: 2601.10427 · v2 · pith:AKYAM3RVnew · submitted 2026-01-15 · ❄️ cond-mat.dis-nn · math-ph· math.MP· math.PR

The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices

Pierre Bousseyroux , Marc Potters This is my paper

Pith reviewed 2026-05-16 14:11 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn math-phmath.MPmath.PR
keywords non-Hermitian random matricesfinite-rank perturbationsoutlier eigenvalueseigenvector statisticsspectral analysisrandom matrix ensembles
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The pith

Finite-rank normal perturbations create outlier eigenvalues outside the bulk spectrum of large rotationally invariant non-Hermitian random matrices.

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

The paper analyzes how adding a finite-rank normal perturbation T to a large rotationally invariant non-Hermitian random matrix A leads to the appearance of isolated outlier eigenvalues. It derives the locations of these outliers, their fluctuation statistics, and the behavior of the corresponding eigenvectors. A sympathetic reader would care because this provides explicit predictions for spectral features in non-Hermitian random matrix models common in physics and machine learning. If true, such models allow reliable identification of dominant eigenvalues without needing to diagonalize the full matrix.

Core claim

We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. We characterize the emergence and fluctuations of outlier eigenvalues in models of the form A + T, where A is a large rotationally invariant non-Hermitian random matrix and T is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.

What carries the argument

The finite-rank normal perturbation T that interacts with the bulk spectrum of A to produce isolated eigenvalues whose positions solve a fixed-point equation involving the resolvent of A.

Load-bearing premise

The matrix A must be rotationally invariant and T must be both normal and finite-rank for the outlier characterization to hold.

What would settle it

For a specific large matrix A drawn from the circular law ensemble and a rank-one normal T with eigenvalue lambda, the largest eigenvalue of A + T should be approximately lambda if |lambda| is large enough, and deviate from the bulk edge otherwise.

read the original abstract

We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-P\'ech\'e (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.

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

0 major / 2 minor

Summary. The manuscript extends the Baik-Ben Arous-Péchè (BBP) framework to finite-rank normal perturbations T of large rotationally invariant non-Hermitian random matrices A. It characterizes the locations, emergence thresholds, and fluctuations of outlier eigenvalues of A + T together with the associated eigenvector behavior, yielding a unified treatment that recovers known Hermitian results as special cases.

Significance. If the derivations hold, the work is significant: it supplies explicit, testable characterizations of outliers and eigenvectors under the stated invariance and normality conditions, thereby closing a gap between Hermitian BBP theory and its non-Hermitian counterparts that appear in applications such as neural-network spectra and open quantum systems. The parameter-free nature of the limiting formulas (once the rotational invariance of A is fixed) is a notable technical strength.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'rotationally invariant' is used without a one-sentence definition; adding the precise invariance assumption (e.g., the joint distribution of entries is invariant under unitary conjugation) would clarify the scope for readers unfamiliar with the non-Hermitian literature.
  2. [Introduction] The manuscript would benefit from an explicit statement, early in the introduction, of the precise moment or tail conditions imposed on the entries of A to guarantee the circular-law convergence that underpins the outlier analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the recognition of the unified framework, and the recommendation of minor revision. No specific major comments appear in the report, so we have no point-by-point revisions to propose at this stage. We remain ready to incorporate any minor suggestions once they are communicated.

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions identified

full rationale

The paper extends the classical BBP framework to characterize outlier eigenvalues and eigenvectors for A + T where A is large rotationally invariant non-Hermitian and T is finite-rank normal. The stated conditions (rotational invariance of A, normality and finite rank of T) are explicit modeling assumptions required for the formulas to hold, not derived quantities. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claim to its inputs appear in the provided abstract or description. The unification of Hermitian and non-Hermitian cases is presented as an extension of prior independent results without internal reduction to fitted inputs or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the rotational invariance of the unperturbed matrix and the normality plus finite rank of the perturbation; these are domain assumptions standard in random-matrix theory but not independently verified in the abstract.

axioms (2)
  • domain assumption The unperturbed matrix A is rotationally invariant
    Stated directly in the abstract as the model class under study.
  • domain assumption The perturbation T is normal and of finite rank
    Required for the outlier characterization to apply.

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Forward citations

Cited by 3 Pith papers

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  1. BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise

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    For a spiked Wigner model with power-law inhomogeneous noise variances, the BBP transition is non-monotonic and inhomogeneous noise can enhance signal detectability.

  2. Quantum many-body operator cascade as a route to chaos

    cond-mat.stat-mech 2026-04 unverdicted novelty 6.0

    Local operators in quantum chaotic systems cascade toward non-local fractal structures whose dimension is tied by unitarity to the decay rate of local correlations, demonstrated exactly in dual-unitary circuits and nu...

  3. Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

    cond-mat.dis-nn 2026-02 unverdicted novelty 6.0

    Spectral boundaries of A + B (A deterministic, B rotationally invariant random non-Hermitian) are given by simple equations depending on the R1 and R2 transforms of B in the large-N limit.

Reference graph

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