Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 42, Issue 4, Pages 1732-1757Publisher
SIAM PUBLICATIONS
DOI: 10.1137/21M1390621
Keywords
symplectic eigenpairs; Williamson's diagonal form; trace minimization; Riemannian optimization; symplectic Stiefel manifold; symmetric positive-definite matrices; positive-definite Hamiltonian matrix
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Funding
- Fonds de la Recherche Scientifique-FNRS
- Fonds Wetenschappelijk Onderzoek-Vlaanderen under EOS project [30468160]
- Vietnam Institute for Advanced Study in Mathematics (VIASM)
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This paper addresses the problem of computing the smallest symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices. It formulates the problem as minimizing a trace cost function over the symplectic Stiefel manifold. The authors propose a numerical procedure for computing the symplectic eigenpairs and discuss the connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices.
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 nonglobal 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 a numerical procedure for computing symplectic eigenpairs 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.
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