Semester 2 / Methods of Mathematics / Multivariable Calculus / Hessian Matrix

Hessian Matrix

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Suppose f:Rn→Rf: \mathbb{R}^n \to \mathbb{R} is a twice partially differentiable function. The Hessian matrix of ff is the matrix of second partial derivatives.

H(f)=D2f=(∂2f∂x12∂2f∂x2∂x1⋯∂2f∂xn∂x1∂2f∂x1∂x2∂2f∂x22⋯∂2f∂xn∂x2⋮⋮⋱⋮∂2f∂x1∂xn∂2f∂x2∂xn⋯∂2f∂xn2)H(f) = D^2 f = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_2 \partial x_1} & \cdots & \frac{\partial^2 f}{\partial x_n \partial x_1} \\ \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_n \partial x_2} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_1 \partial x_n} & \frac{\partial^2 f}{\partial x_2 \partial x_n} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}

At a point x∈Rnx \in \mathbb{R}^n is the n×nn \times n matrix of second partial derivatives of ff at xx. Denoted by H(f)(x)H(f)(x).

Written by September 29, 2026 1 min read
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