Brockett cost function for symplectic eigenvalues

Nguyen Thanh Son

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arXiv · 2506.07560

Brockett cost function for symplectic eigenvalues

Nguyen Thanh Son

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Abstract

The sum of symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem can be computed via minimization of a trace cost function under the symplecticity constraint. Optimal solutions to this problem only offer a symplectic basis for the symplectic eigenspace corresponding to the sought symplectic eigenvalues. In this note, we introduce a Brockett cost function and investigate its properties and the connection with the symplectic eigenvalues and eigenvectors of the considered matrix. Specifically, we prove that any stationary point consists of symplectic eigenvectors, characterize the saddle points and global minimizers based on which the trace minimization theorem for the symplectic eigenvalues is re-established, and the nonexistence of local nonglobal minimizers is justified.

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BibTeX RIS

Nguyen Thanh Son. 2026-09-14. Brockett cost function for symplectic eigenvalues. https://arxiv.org/abs/2506.07560

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