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arxiv: 2605.26317 · v1 · pith:HRDMWGRLnew · submitted 2026-05-25 · 🧮 math.NA · cs.NA

A Jacobi-like algorithm for normal matrices by the skew-symmetric part

Simon Mataigne , P.-A. Absil This is my paper

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classification 🧮 math.NA cs.NA
keywords Jacobi algorithmnormal matriceseigenvalue computationskew-symmetric matricesPaardekooper methodorthogonal matricesnumerical linear algebra
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The pith

Jacobi-like algorithm for real normal matrices gains speed by applying Paardekooper's method to the skew-symmetric part

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

The paper presents a Jacobi-like algorithm for the eigenvalues and optional eigenvectors of real normal matrices. It incorporates Paardekooper's method on the skew-symmetric component to reduce work when most eigenvalues are complex. This setting covers random orthogonal matrices that appear in statistics on manifolds, where the new procedure runs faster than prior Jacobi-like algorithms. The final section supplies closed-form expressions for the nearest symmetric skew-Hamiltonian matrix and the nearest ortho-symplectic matrix, problems that arise directly in the algorithm's construction and analysis.

Core claim

A Jacobi-like iteration for real normal matrices that routes the skew-symmetric part through Paardekooper's method computes eigenvalues more rapidly than existing Jacobi-like schemes when most eigenvalues are complex, and yields explicit formulas for the nearest symmetric skew-Hamiltonian and nearest ortho-symplectic matrices.

What carries the argument

Integration of Paardekooper's skew-symmetric eigenvalue routine into a Jacobi-like sweep for normal matrices

Load-bearing premise

That routing the skew-symmetric part through Paardekooper's method inside the Jacobi iteration produces a net reduction in arithmetic work for matrices whose eigenvalues are mostly complex.

What would settle it

A direct timing comparison on a collection of random orthogonal matrices in which the new algorithm requires at least as many floating-point operations or wall-clock time as the fastest existing Jacobi-like method.

read the original abstract

We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The method is most efficient for matrices where most eigenvalues are complex, such as random orthogonal matrices arising in the context of statistics on manifolds. In this case, the method is faster than the other Jacobi-like algorithms. In the last section of this paper, we also give explicit formulas for the nearest symmetric skew-Hamiltonian and the nearest ortho-symplectic matrix. These problems arise in the design and the analysis of the algorithm.

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 manuscript presents a Jacobi-like algorithm for computing the eigenvalues (and optionally eigenvectors) of real normal matrices. It integrates Paardekooper's method for the skew-symmetric part to obtain a computational advantage, claiming superior speed over other Jacobi-like methods when most eigenvalues are complex (e.g., random orthogonal matrices arising in manifold statistics). Explicit formulas are also derived for the nearest symmetric skew-Hamiltonian matrix and the nearest ortho-symplectic matrix.

Significance. If the speed advantage is confirmed by reproducible timings and analysis, the algorithm would supply a targeted, practical tool for eigenvalue problems on normal matrices with complex spectra. The explicit nearest-matrix formulas may additionally serve as independent contributions for structured-matrix approximation tasks.

major comments (1)
  1. [Abstract] Abstract: the central claim that the method 'is faster than the other Jacobi-like algorithms' for matrices with mostly complex eigenvalues is unsupported by any numerical timings, operation counts, or complexity comparison in the visible text; this performance assertion is load-bearing and cannot be assessed from the given material.
minor comments (1)
  1. The abstract refers to 'the last section of this paper' for the nearest-matrix formulas without providing section numbers or indicating how these formulas are used in the algorithm's design or analysis.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript's significance and for identifying the need to substantiate the performance claim. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the method 'is faster than the other Jacobi-like algorithms' for matrices with mostly complex eigenvalues is unsupported by any numerical timings, operation counts, or complexity comparison in the visible text; this performance assertion is load-bearing and cannot be assessed from the given material.

    Authors: We agree that the performance assertion in the abstract requires explicit support and cannot be left unsubstantiated. The advantage stems from integrating Paardekooper's method on the skew-symmetric part, which avoids full complex arithmetic per rotation when eigenvalues are complex conjugate pairs; this yields a lower operation count per sweep compared with standard Jacobi-like methods that treat all entries uniformly. In the revised manuscript we will add a dedicated numerical section with reproducible timings on random orthogonal matrices (and other normal matrices with predominantly complex spectra), together with an operation-count analysis contrasting the per-iteration cost against the methods cited in the introduction. The claim will be qualified accordingly and moved from the abstract to the results section once the evidence is presented. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper integrates Paardekooper's external method for skew-symmetric matrices into a Jacobi-like scheme for normal matrices, with the speed advantage presented as an empirical observation on matrices with mostly complex eigenvalues (e.g., random orthogonal matrices). No load-bearing derivations, self-citations, or fitted inputs reduce to the target result by construction. The additional explicit formulas for nearest skew-Hamiltonian/ortho-symplectic matrices are described as arising in design/analysis and are not foundational to the performance claim. The approach is independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work relies on standard numerical linear algebra background and an external cited method.

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Reference graph

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