# Positive semidefinite cone

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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]] | ||

- | + | 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 |

- | each halfspace having partial boundary containing the origin in an isomorphic subspace. Hence | + | <strong>(</strong>as in figure<strong>)</strong>, |

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

- | + | ||

- | The only symmetric positive semidefinite matrix having | + | Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices. |

- | + | ||

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

+ | |||

+ | The only symmetric positive semidefinite matrix having all <math>0</math> 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 <math>T</math> relating matrix space to vector space: | ||

+ | <ul> | ||

+ | <li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> ||<math>x - y</math>||<sub>F</sub> = ||<math>Tx - Ty</math>||<sub>2</sub> | ||

+ | <strong>(</strong>distance between matrices <math>=</math> distance between vectorized matrices<strong>)</strong>. | ||

+ | <li>In one dimension, 1×1 symmetric matrices, the nonnegative ray is a circular cone. | ||

+ | </ul> | ||

+ | |||

+ | [http://meboo.convexoptimization.com/access.html Read more...] |

## Current revision

*"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.