definiteness of the Hessian
/ Hessian: HESS-ee-uhn /
The Hessian matrix collects all the second partial derivatives of a function and so describes the curvature of its surface in every direction at once. Definiteness is the single property of that matrix that tells you whether the curvature is bowl-up, bowl-down, or mixed. It is the precise meaning behind the words minimum, maximum, and saddle.
A symmetric matrix H is called positive definite if the quadratic form v transpose times H times v is strictly positive for every nonzero direction v — meaning the surface curves upward no matter which way you head. It is negative definite if that form is always negative (curves down everywhere), and indefinite if the form is positive for some directions and negative for others (the saddle). The cleanest test is the eigenvalues: positive definite means all eigenvalues are positive, negative definite means all negative, indefinite means a mix of signs. An equivalent test in low dimensions checks the signs of the leading principal minors (Sylvester's criterion). When some eigenvalues are zero the matrix is only semidefinite, and the second-order test cannot decide.
Definiteness is what makes the second-derivative test work, but its reach is far wider. A function whose Hessian is positive semidefinite everywhere is convex, the gold standard for optimization because every stationary point is then a global minimum. Definiteness governs the stability of equilibria in mechanics (a positive definite stiffness matrix means a stable rest position), the well-posedness of least-squares problems, and whether a covariance matrix in statistics is legitimate. It is the bridge between calculus and linear algebra.
The matrix [2, 1; 1, 2] has eigenvalues 3 and 1, both positive, so it is positive definite — a function with this Hessian sits at a local minimum. The matrix [1, 0; 0, -1] has eigenvalues 1 and -1, mixed signs, so it is indefinite — a saddle.
Eigenvalue signs read off definiteness directly: all-positive is a minimum, mixed is a saddle.
Positive diagonal entries alone do not guarantee positive definiteness — the off-diagonal couplings matter. [1, 2; 2, 1] has positive diagonal but eigenvalues 3 and -1, so it is indefinite.