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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.
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