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arxiv: 2606.20432 · v1 · pith:WWB5GL5Inew · submitted 2026-06-18 · 🧮 math.AG · math.RA· quant-ph

Eigenvector Varieties

Sandra Di Rocco , Bernd Sturmfels , Svala Sverrisd\'ottir This is my paper

Pith reviewed 2026-06-26 15:14 UTC · model grok-4.3

classification 🧮 math.AG math.RAquant-ph
keywords eigenvector varietylinear spaces of matricesLie algebrasquantum Hamiltoniansprojective varietiesalgebraic geometry
0
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The pith

Any linear space of square matrices determines an eigenvector variety consisting of the eigenvectors of its matrices.

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

The paper defines an eigenvector variety for any linear space of square matrices as the set of points that are eigenvectors for matrices in that space. It conducts a systematic study of these varieties, with special attention to those arising from Lie algebras and from Hamiltonians of quantum systems. A sympathetic reader would care because this turns a collection of matrices into a single geometric object in projective space whose structure can be analyzed with the tools of algebraic geometry.

Core claim

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space.

What carries the argument

The eigenvector variety of a linear space of matrices, the projective variety whose points are the eigenvectors (common or individual) of the matrices in the space.

If this is right

  • The eigenvector variety is an algebraic variety whose dimension and degree are determined by the linear space.
  • Geometric invariants of the variety reflect algebraic properties of the original linear space of matrices.
  • For Lie algebras the variety encodes geometric information about joint eigenvectors or orbits.
  • For quantum Hamiltonians the variety gives a projective-geometric model of the relevant eigenspaces.

Where Pith is reading between the lines

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

  • Elimination or resultant techniques could be used to compute explicit equations for these varieties in low dimensions.
  • The construction may relate eigenvector varieties to other classical objects such as commuting varieties or determinantal varieties.
  • The same definition could be applied to linear spaces over other fields or to rectangular matrices to produce analogous varieties.

Load-bearing premise

The eigenvectors of matrices belonging to an arbitrary linear space always form an algebraic variety in projective space.

What would settle it

An explicit example of a linear space of square matrices whose eigenvectors do not form a closed algebraic subset of projective space.

read the original abstract

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

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 defines the eigenvector variety of a linear space L of square matrices as the closed subscheme of projective space cut out by the condition that every matrix in L has a given vector as eigenvector. It announces a systematic study of these varieties, with emphasis on the cases where L arises as a Lie algebra or as a space of Hamiltonians for quantum systems.

Significance. The construction supplies a natural algebraic-geometric object attached to any linear space of matrices; the explicit description via 2×2 minors of the matrices [v | A_i v] shows that the locus is always a projective scheme without further hypotheses. If the subsequent study yields concrete classifications, dimension formulas, or applications to representation theory and quantum control, the framework could be useful for organizing existing results on common eigenspaces.

minor comments (3)
  1. The introduction should state the precise definition (including the ambient projective space and the ideal generated by the minors) before announcing the systematic study, so that the reader can immediately verify the claim that the locus is always a variety.
  2. When the focus turns to Lie algebras and quantum Hamiltonians, explicit low-dimensional examples (e.g., sl(2) or a two-qubit Hamiltonian space) with computed equations or Hilbert polynomials would make the claimed systematic study more concrete.
  3. Notation for the eigenvector variety (e.g., E(L) or V(L)) should be fixed early and used consistently; the abstract uses none.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on eigenvector varieties. The recommendation of minor revision is noted; however, the report contains no specific major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity: eigenvector variety is defined by explicit equations

full rationale

The paper introduces the eigenvector variety as the projective locus of vectors v such that A v lies in span(v) for all A in a linear space L. This locus is cut out by the 2x2 minors of the matrices [v | A_i v] for a basis of L; each minor is a homogeneous quadratic, so the set is a closed subscheme by the definition of projective varieties. No derivation step reduces this construction to fitted parameters, self-citations, or prior results by the same authors. The subsequent study of examples (Lie algebras, Hamiltonians) proceeds from this explicit algebraic definition without circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities beyond the definitional introduction of the eigenvector variety itself are stated.

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Reference graph

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