We define

• the subspace/submodule of symmetric matrices
• the cone of positive-semidefinite matrices (with ambient = the full matrix space; affine_hull = the above subspace), implementing the ABC for convex sets (ABC for convex sets #31919)
• the cone of positive-definite matrices as its relative_interior.
• additive semigroups/monoids of positive-(semi)definite matrices as intersection of the above with a MatrixSpace

Related:

• sage.tensor.modules: Add SymPowerFreeModule #30095 sage.tensor.modules: Add SymPowerFreeModule
• Category InnerProductSpaces #30218 Category InnerProductSpaces
• ManifoldSubset and ChartFunction objects corresponding to a CVXPY expressions #31981 ManifoldSubset corresponding to a CVXPY Leaf (Variable/Parameter)
• sage.manifolds: Connect to Pymanopt #30495 sage.manifolds: Connect to Pymanopt

Random references:

• Jean-Baptiste Hiriart-Urruty, Jérôme Malick, A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World, https://hal.archives-ouvertes.fr/hal-00651605/document
• Yuhan Jiang & Bernd Sturmfels, Bad projections of the PSD cone, 2021, https://link.springer.com/article/10.1007/s13348-021-00319-4
• Lek-Heng Lim , Rodolphe Sepulchre, Ke Ye, Geometric Distance Between Positive Definite Matrices of Different Dimensions, https://www.stat.uchicago.edu/~lekheng/work/ellipsoids.pdf
• James Saunderson, Limitations on the expressive power of convex cones without long chains of faces, https://arxiv.org/abs/1902.06401v2

CC: @dimpase @yuan-zhou @orlitzky @kliem

Component: geometry

Author: Matthias Koeppe

Branch/Commit: u/mkoeppe/cone_of_positive_semidefinite_matrices @ 7e16b79

Issue created by migration from https://trac.sagemath.org/ticket/30172