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arxiv: 1907.03368 · v1 · pith:H4OZBX4Wnew · submitted 2019-07-07 · 🧮 math.FA · math.AG· math.MG

Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms

Jorge Antezana , Eduardo Ghiglioni , Demetrio Stojanoff This is my paper

Pith reviewed 2026-05-25 00:55 UTC · model grok-4.3

classification 🧮 math.FA math.AGmath.MG
keywords minimal curvesFinsler metricunitary groupspectral normtrace normgeodesic convexityHermitian matrices
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The pith

Minimal curves between unitaries are unique exactly when their relative spectrum is contained in a symmetric pair on the unit circle.

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

The paper provides a complete description of all length-minimizing curves in the unitary group U(n) equipped with the bi-invariant Finsler metric coming from the spectral norm. It shows that infinitely many such curves exist in general between two points but supplies an explicit list of them. Uniqueness holds if and only if the spectrum of U*V lies inside a set of the form {e^{i θ}, e^{-i θ}} for some θ. The same methods yield a characterization of minimal curves in the positive invertible matrices under the trace norm, obtained by reducing to the Hermitian case, plus convexity statements for the sets of intermediate points on those curves.

Core claim

In U(n) with the spectral-norm Finsler metric there is a unique minimal-length curve between U and V if and only if the spectrum of U*V is contained in {e^{i θ}, e^{-i θ}} for some θ; a full description of all minimal curves is given in the general case. For Gl(n)+ with the trace-norm metric the minimal curves are characterized by first describing the minimal curves between Hermitian matrices, which in turn produces minimal paths in U(n) under the trace norm. The set of intermediate points between two unitaries is geodesically convex whenever the spectral distance is less than 1, and the corresponding set in Gl(n)+ is always geodesically convex for any unitarily invariant norm.

What carries the argument

The spectrum of U*V, which determines whether the minimal curve in the bi-invariant spectral-norm Finsler metric on U(n) is unique.

If this is right

  • When the spectrum of U*V satisfies the two-point condition the shortest curve is the only one.
  • Minimal curves in Gl(n)+ are obtained from the explicit description of minimal curves joining Hermitian matrices.
  • The intermediate-point set is geodesically convex whenever ||U-V||_sp < 1 for the spectral norm and for every unitarily invariant norm in Gl(n)+.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same spectral criterion may classify uniqueness of minimal curves for other bi-invariant Finsler metrics on compact Lie groups.
  • The explicit list of minimal curves supplies a practical way to compute distances in U(n) for any pair satisfying the condition.
  • Geodesic convexity of the intermediate set implies that small perturbations of the endpoints keep the minimal curves inside a convex region.

Load-bearing premise

The length of a curve is obtained by integrating the chosen norm of its velocity after translating back to the Lie algebra via the bi-invariance of the metric.

What would settle it

Choose concrete unitary matrices U and V such that U*V has at least three distinct eigenvalues that do not fit inside any set {e^{i θ}, e^{-i θ}}, then compute whether more than one distinct curve between them attains the same infimal length.

read the original abstract

Consider the Lie group of n x n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm, ||X||_U = ||U*X||_{sp} = ||X||_{sp} for any X tangent to a unitary operator U. Given two points in U(n), in general there exists infinitely many curves of minimal length. The aim of this paper is to provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between U and V if and only if the spectrum of U*V is contained in a set of the form \{e^{i \theta}, e^{-i \theta}\} for some \theta \in [0, \infty). Similar studies are done for the Grassmann manifolds. Now consider the cone of n x n positive invertible matrices Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ||X||_{1, A} = ||A^{-1/2}XA^{-1/2}||_1 for any X tangent to A \in Gl(n)+. In this context, given two points A,B \in Gl(n)+ there exists infinitely many curves of minimal length. In order to provide a complete description of such curves, we provide a characterization of the minimal curves joining two Hermitian matrices X, Y \in H(n). As a consequence of the last description, we provide a way to construct minimal paths in the group of unitary matrices U(n) endowed with the bi-invariant Finsler metric ||X||_{1, U} = ||U*X||_{1} = ||X||_{1} for any X tangent to U \in U(n). We also study the set of intermediate points in all the previous contexts. Between two given unitary matrices U and V we prove that this set is geodesically convex provided ||U - V||_{sp} < 1. In Gl(n)+ this set is geodesically convex for every unitarily invariant norm.

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 / 3 minor

Summary. The manuscript claims to give a complete description of length-minimizing curves for the bi-invariant Finsler metric induced by the spectral norm on U(n), from which it derives that a unique minimal curve between U and V exists if and only if spec(U*V) is contained in a set of the form {e^{iθ}, e^{-iθ}} for some θ ≥ 0. Analogous results are stated for the trace-norm metric on GL(n)+, including a characterization of minimal curves between Hermitian matrices X, Y ∈ H(n) that is then used to construct minimal paths in U(n), together with geodesic-convexity statements for the set of intermediate points (under the condition ||U−V||_sp < 1 on U(n) and for every unitarily invariant norm on GL(n)+). Similar studies are announced for Grassmann manifolds.

Significance. If the characterizations hold, the work supplies explicit if-and-only-if uniqueness criteria and convexity results for bi-invariant Finsler metrics on classical matrix groups; such statements are useful for geodesic analysis and could inform numerical methods on unitary and positive-definite manifolds. The reduction from GL(n)+ to Hermitian matrices and the spectrum-based uniqueness condition are concrete contributions when rigorously established.

minor comments (3)
  1. The abstract asserts complete characterizations and an if-and-only-if uniqueness statement; the main text should make the key reduction steps and supporting lemmas for the spectrum condition explicit (e.g., the precise section deriving the iff statement from the curve description).
  2. The statement that the set of intermediate points is geodesically convex for ||U−V||_sp < 1 on U(n) is load-bearing for applications; a brief indication of the argument (or reference to the relevant proposition) would improve readability.
  3. Notation for the norms (||·||_sp, ||·||_1, ||·||_{1,A}) is introduced in the abstract but should be recalled with a short reminder at the beginning of each main section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript, the positive significance assessment, and the recommendation of minor revision. No specific major comments appear in the provided report, so we have nothing to address point-by-point at this stage. We remain available to incorporate any minor suggestions from the editor or referee.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives a complete characterization of length-minimizing curves for the bi-invariant spectral-norm Finsler metric on U(n) and the trace-norm metric on Gl(n)+, then obtains the spectrum condition for uniqueness as a direct consequence. No step reduces by definition to its own inputs, no parameter is fitted to a subset and renamed as a prediction, and no load-bearing premise rests on a self-citation chain. The reductions to Hermitian matrices and geodesic-convexity statements are independent of the target uniqueness result and rely on standard Lie-group and Finsler-geometry techniques that remain externally verifiable. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions about Lie groups and unitarily invariant norms; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The spectral norm defines a bi-invariant Finsler metric on the Lie group U(n).
    Explicitly used to set up the first part of the study.
  • domain assumption The trace norm defines a bi-invariant Finsler metric on the cone Gl(n)+.
    Explicitly used to set up the second part of the study.

pith-pipeline@v0.9.0 · 5939 in / 1279 out tokens · 36225 ms · 2026-05-25T00:55:40.493050+00:00 · methodology

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