Semidefinite programming (SDP) optimizes a linear objective over the intersection of an affine subspace with the positive semidefinite (PSD) cone — the set of matrices X with all eigenvalues ≥ 0. This cone generalizes the non-negative orthant of LP and is self-dual. SDPs arise in control theory, quantum information, graph theory (MAX-CUT relaxations via Goemans-Williamson achieve 0.878 approximation), and sum-of-squares proofs. The ellipsoid in eigenspace visualized here shows how the PSD cone is a curved boundary — feasible matrices correspond to ellipsoids that fit inside the unit ball, parameterized by their Cholesky factor.