An elementary proof of the symplectic spectral theorem

TL;DR

This paper provides an elementary proof of a spectral theorem for self-adjoint operators on finite-dimensional symplectic vector spaces, extending classical results to a symplectic setting with explicit basis constructions.

Contribution

It introduces a symplectic spectral theorem with a simple proof and describes the decomposition of operators via a polarization, including explicit matrix forms when eigenvalues are in the base field.

Findings

Operators decompose according to a polarization

Existence of Darboux basis with block-diagonal form

Explicit Jordan form for operators with eigenvalues in the base field

Abstract

The classical spectral theorem completely describes self-adjoint operators on finite dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a lagrangian subspace and its dual on a complementary lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exist a Darboux basis such that the matrix representation of the operator is two-by-two blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one.