# Positive semidefinite cone

### From Wikimization

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[[Image:psdcone.jpg|right|positive semidefinite cone is a circular cone in 3D]] | [[Image:psdcone.jpg|right|positive semidefinite cone is a circular cone in 3D]] | ||

- | <br> | ||

The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone: | 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 | + | It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix |

+ | <strong>(</strong>as in figure<strong>)</strong>, | ||

each halfspace having partial boundary containing the origin in an isomorphic subspace. | each halfspace having partial boundary containing the origin in an isomorphic subspace. | ||

## Revision as of 22:17, 7 July 2011

*"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 zero eigenvalues resides at the origin.

In low dimension the positive semidefinite cone is a circular cone because there is 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 equals distance between vectorized matrices**)**.
In one dimension, the nonnegative ray is a circular cone.