WHAT IS...a Spectrahedron?
Cynthia Vinzant · Notices of the American Mathematical Society · 2014
A spectrahedron is a convex set that appears in a range of applications.Introduced in [3], the name joins "spectra", evoking the eigenvalues of a matrix, with "hedron", suggesting that spectrahedra generalize convex polyhedra.First we need to recall some linear algebra.All the eigenvalues of a real symmetric matrix are real, and if these eigenvalues are all nonnegative then the matrix is positive semidefinite.The set of positive semidefinite matrices is a convex cone in the vector space of real symmetric matrices.A spectrahedron is the intersection of an affine linear space with this convex cone of matrices.An n-dimensional affine linear space of real symmetric matrices can be parameterized byranges over R n , where A 0 , . . ., A n are real symmetric matrices.This identifies our spectrahedron with the set of x in R n for which the matrix A(x) is positive semidefinite.This condition, denoted A(x) 0, is commonly known as a linear matrix inequality.For example, we can write the cylinder