"The cone of positive semidefinite matrices studied in this section is arguably the most important of all non-polyhedral cones whose facial structure we completely understand."

The set of all symmetric positive semidefinite matrices of particular dimension  is called the positive semidefinite cone: It can be formed by the intersection of an infinite number of halfspaces in vectorized variable A, each halfspace having partial boundary containing the origin in an isomorphic subspace. Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.

The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices (having at least one 0 eigenvalue) reside on the cone boundary.

The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.

In low dimension the positive semidefinite cone is a circular cone.

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