2021/01/01 by Nguyen Thanh Son, P.-A. Absil, P. -A. Absil +2 · 20 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Algebra over a field #Applied mathematics #Control and Stability of Dynamical Systems #Eigenvalues and eigenvectors #Geometry #Hamiltonian (control theory) #Manifold (fluid mechanics) #Mathematical optimization #Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Positive-definite matrix #Pure mathematics #Quadratic programming #Riemannian manifold #Saddle point #Semidefinite embedding #Stiefel manifold #Symplectic geometry #Symplectic manifold #Symplectic matrix #Symplectic representation #Symplectic vector space #TRACE (psycholinguistics) #math.OC #math.SP #msc:15A15 #msc:15A18 #msc:70G45
paper · pdf · open access · doi:10.1137/21m1390621
published in SIAM Journal on Matrix Analysis and Applications 42(4), 1732-1757 (Society for Industrial and Applied Mathematics) · 24 pages, 2 figures
openalex publication_date 2021/01/01 · arxiv created 2021/01/07 · arxiv updated 2022/01/06 · openalex created_date 2022/01/25 · openalex updated_date 2026/07/31
We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented