What is... a spectrahedron

A spectrahedron is a convex set that appears in a range of applications. Introduced in [3], the name “spectra” is used because its definition involves the eigenvalues of a matrix and “-hedron” because these sets generalize 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 non-negative 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 affinelinear space with this convex cone of matrices. An n-dimensional affine-linear space of real symmetric matrices can be parametrized by