A skew-symmetric matrix of odd size has eigenvalue 0

Proved

UnQuantumMechanics.skew_symm_odd_zero_eigenvalue

by Lucas · Oct 2, 2026 · Mathlib 0df444a (Lean v4.33.1)

hamiltonian-mechanicslinear-algebramathematical-physics

Let $\nu$ be odd and let $J$ be a real $\nu\times\nu$ matrix with $J^T = -J$ . Then $0$ is an eigenvalue of $J$ : there is a nonzero $v \in \mathbb R^\nu$ with

J v = 0 .

In the paper this is the reason why a full-rank structure matrix requires an even number $\nu = 2n$ of dynamical variables.

Preamble

import Mathlib
import Definitions.Def_UnQM_phase_space

open Matrix

Formal statement

namespace UnQuantumMechanics

theorem skew_symm_odd_zero_eigenvalue {ν : ℕ} (hν : Odd ν) (J : Matrix (Fin ν) (Fin ν) ℝ)
    (hJ : Jᵀ = -J) :
    Module.End.HasEigenvalue (Matrix.toLin' J) 0 := by sorry

end UnQuantumMechanics

Source

C. Baumgarten, How to (Un-) Quantum Mechanics, arXiv:1810.06981v2 [physics.gen-ph] (v2 dated June 10, 2025), https://arxiv.org/abs/1810.06981, Sec. IV, p. 13, paragraph after Eq. (28)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent that drafted the statements; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement, at the explicit direction of the proposal owner. The author knew the intended meaning and the source text while writing it, so it must not be mistaken for a blind, independent audit. An independent blind read-back is still recommended before submission.

For every odd natural number $\nu$ and every real $\nu\times\nu$ matrix $J$ with $J^T = -J$ , the number $0$ is an eigenvalue of the linear map $\mathbb R^\nu \to \mathbb R^\nu$ , $v \mapsto Jv$ ; that is, there exists a nonzero real vector $v$ with $Jv = 0$ . Since $\nu$ is odd, $\nu \ge 1$ , so the space is nonzero.

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

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