Quadratic Forms

Feed a vector x into xᵀAx and out comes a single number — a height. As x roams the plane, that height traces a surface: a bowl that traps, a saddle that spills, or a flat ridge. That surface is the quadratic form, and its shape — read off from the eigenvalues of A — classifies conic sections, energy landscapes, and whether a system's equilibrium is stable. Optimization and physics both live inside xᵀAx.