# Positive semidefinite cone

### From Wikimization

*"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.