The Hessian matrix H(f) of a function f: Rⁿ → R is the n×n matrix of all second partial derivatives. For f(x,y): [ fxx fxy ] [ fyx fyy ] Definition: [ ∂²f/∂x₁² ∂²f/∂x₁∂x₂ ... ∂²f/∂x₁∂xₙ ] [ ∂²f/∂x₂∂x₁ ∂²f/∂x₂² ... ∂²f/∂x₂∂xₙ ] H(f) = [ : : : ] [ ∂²f/∂xₙ∂x₁ ∂²f/∂xₙ∂x₂ ... ∂²f/∂xₙ² ] In compact form: [H(f)]ᵢⱼ = ∂²f / ∂xᵢ∂xⱼ Named after German mathematician Otto Hesse (1811–1874). Used to study curvature, convexity, and classify critical points in multivariable calculus & optimization.