Spectrahedron Explained

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

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

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

where

S ^{n} _{+} is the set of positive semidefinite matrices and \operatorname{Tr}(X) is the trace of the matrix 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.

Notes and References

1. .
2. Scheiderer. C.. 2018-01-01. Spectrahedral Shadows. SIAM Journal on Applied Algebra and Geometry. 2. 26–44. 10.1137/17m1118981. free.
3. Book: Gärtner, Bernd. Approximation Algorithms and Semidefinite Programming. limited. Matousek. Jiri. Springer Science and Business Media. 2012. 978-3642220159. 76.