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 a linear affine subspace.

Spectrahedra are the solution spaces of semidefinite programs.

An example of a spectrahedron is the similarly named spectahedron, defined as

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

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

...
Wikipedia