Spectrahedron

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of n × n positive semidefinite matrices forms a convex cone in R^{n × n}, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.

A spectrahedron

Spectrahedra are the feasible regions of semidefinite programs.^[1] The images of spectrahedra under linear or affine transformations are called projected spectrahedra or spectrahedral shadows. Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false.^[2]

An example of a spectrahedron is the spectraplex, defined as

${\displaystyle \mathrm {Spect} _{n}=\{X\in \mathbf {S} _{+}^{n}\mid \operatorname {Tr} (X)=1\}}$,

where ${\displaystyle \mathbf {S} _{+}^{n}}$ is the set of n × n positive semidefinite matrices and ${\displaystyle \operatorname {Tr} (X)}$ is the trace of the matrix ${\displaystyle X}$.^[3] The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.

See also

• N-ellipse - a special case of spectrahedra.