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Hesse Eigenvalues: Curvature on the Principal Axes

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The Hesse matrix is symmetric, so it diagonalises in an orthonormal basis: its eigenvectors are the principal directions of curvature and its eigenvalues are the curvatures along them. Both eigenvalues positive — minimum; both negative — maximum; mixed signs — saddle. The discriminant just tracks the product λ₁ λ₂ = det H.

Hesse matrixeigenvalueseigenvectorsprincipal curvaturediagonalisationspectral theorem
ForutsetterThe Second-Derivative Test: A Discriminant DecidesDiagonalisation: Eigenvectors as Natural Axes
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