Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric

  title={Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric},
  author={Bin Gao and Nguyen Thanh Son and Pierre-Antoine Absil and Tatjana Stykel},
The symplectic Stiefel manifold, denoted by Sp(2p, 2n), is the set of linear symplectic maps between the standard symplectic spaces R and R. When p = n, it reduces to the well-known set of 2n × 2n symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space R. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel… 
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Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices
  • S. Fiori
  • Mathematics
    SIAM J. Matrix Anal. Appl.
  • 2011
The minimal-distance problem investigated in this paper relies on a suitable notion of distance—induced by the Frobenius norm—as opposed to the natural pseudodistance that corresponds to the pseudo-Riemannian metric that the real-symplectic group is endowed with.
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Abstract Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist n positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb {R}^{2n}$
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If A is a 2n × 2n real positive definite matrix, then there exists a symplectic matrix M such that MTAM=DOOD where D = diag(d1(A), …, dn(A)) is a diagonal matrix with positive diagonal entries, which