Positive Semidefinite Cone | 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."
The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices (having at least one 0 eigenvalue) reside on the cone boundary. In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism T relating matrix space to vector space: For a 2×2 symmetric matrix, T is obtained by scaling the β coordinate by √2 (as in figure). This linear bijective transformation T preserves distance between two points in each respective space; i.e., ||x - y||F = ||Tx - Ty||2 (distance between matrices equals distance between vectorized matrices). In one dimension, the nonnegative ray is a circular cone. |






