11 - Hessian Matrix on Classifying Stationary Points

Hessian Matrix $H$

For a function $f(x_1,\cdots ,x_n)$ of $n$ variables with continuous partial derivatives to second order then

It is a $n\times n$ symmetric matrix with entry $\frac{{\partial }^{2}y}{\partial x_i\partial x_j}$ at the $i$th row and $j$th column

Hence

$$\mathbf{H}(x_1,\cdot ,x_n)=\left(\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial x_1^2}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \cdots & \frac{{\partial }^{2}f}{\partial x_n^2}\end{array}\right)$$

Hessian Matrix on Classifying Stationary Points

Suppose $f(\mathbf{x})$ is a function with $n$ variables so

$$\mathbf{x}=(x_1\cdots ,x_n)$$

which is defined on an open subset $U\subset {\mathbb{R}}^n$ which has

• continuous partial derivatives up to the second order

Let $\mathbf{a}=(a_1,\cdots ,a_n)$ be a critical point such that $\nabla f(\mathbf{a})=0$

1. If the Hessian Matrix $\mathbf{H}(\mathbf{a})$ is positive definite then

$$\mathbf{a}\text{ is a local mininum of }f$$

2. If the Hessian Matrix $\mathbf{H}(\mathbf{a})$ is negative definite then

$$\mathbf{a}\text{ is a local maximum of }f$$