Positive semidefinite cone
"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." Alexander Barvinok
The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone:
It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix (as in figure), 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 eigenvalue) reside on the cone boundary.
The only symmetric positive semidefinite matrix having all eigenvalues resides at the origin.
In low dimension, the positive semidefinite cone is shown to be a circular cone by way of an isometric isomorphism relating matrix space to vector space:
- For a 2×2 symmetric matrix, is obtained by scaling the ß coordinate by √2 (as in figure). This linear bijective transformation preserves distance between two points in each respective space; i.e., ||||F = ||||2 (distance between matrices distance between vectorized matrices).
- In one dimension, 1×1 symmetric matrices, the nonnegative ray is a circular cone.