Hessian Matrix

by W. B. Meitei, PhD

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. Suppose f(x₁, x₂, …, x_n) is a scalar function of n variables, then the Hessian matrix H of f is an n×n matrix defined as,

\[H(f) = \begin{bmatrix} \frac{\partial^{2}f}{\partial x_{1}^{2}} & \frac{\partial^{2}f}{\partial x_{1}\partial x_{2}} & \ldots & \frac{\partial^{2}f}{\partial x_{1}\partial x_{n}} \\ \frac{\partial^{2}f}{\partial x_{2}\partial x_{1}} & \frac{\partial^{2}f}{\partial x_{2}^{2}} & \ldots & \frac{\partial^{2}f}{\partial x_{2}\partial x_{n}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^{2}f}{\partial x_{n}\partial x_{1}} & \frac{\partial^{2}f}{\partial x_{n}\partial x_{2}} & \ldots & \frac{\partial^{2}f}{\partial x_{n}^{2}} \end{bmatrix}\]

 

Note:

• Symmetry: If all second partial derivatives are continuous, the Hessian is symmetric, i.e.,

\[\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}} = \frac{\partial^{2}f}{\partial x_{j}\partial x_{i}}\]

• Critical points and convexity: The Hessian is used to determine the type of critical points (minima, maxima, saddle points). For a twice differentiable function,
• Positive definite Hessian at a point → local minimum.
• Negative definite Hessian at a point → local maximum.
• Indefinite Hessian → Saddle point. 

Suggested Citation: Meitei, W. B. (2025). Hessian Matrix. WBM STATS.