# Positive semidefinite cone

### From Wikimization

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The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin. | The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin. | ||

- | In low dimension, the positive semidefinite cone | + | 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> | <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> | <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> |

## 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.