# Hessian Matrices

We are about to look at a method of finding extreme values for multivariable functions. We will first need to define what is known as the Hessian Matrix (sometimes simply referred to as just the “Hessian”) of a multivariable function.

Definition: Let $\mathbf{x} = (x_1, x_2, …, x_n)$ and let $z = f(x_1, x_2, …, x_n) = f(\mathbf{x})$ be an $n$ variable real-valued function whose second partial derivatives exist. Then the Hessian Matrix of $f$ is the $n \times n$ matrix of second partial derivatives of $f$ denoted $\mathcal H (\mathbf{x}) = \begin{bmatrix} f_{11} (\mathbf{x}) & f_{12} (\mathbf{x}) & \cdots & f_{1n} (\mathbf{x})\\ f_{21} (\mathbf{x}) & f_{22} (\mathbf{x}) & \cdots & f_{2n} (\mathbf{x})\\ \vdots & \vdots & \ddots & \vdots \\ f_{n1} (\mathbf{x}) & f_{n2} (\mathbf{x}) & \cdots & f_{nn} (\mathbf{x}) \end{bmatrix}$. |

For a two variable function $z = f(x, y)$ we have that the Hessian Matrix of $f$ is:

(1)

For a three variable function $w = f(x, y, z)$ we have that the Hessian Matrix of $f$ is:

(2)

Recall from the Clairaut’s Theorem on Higher Order Partial Derivatives page that if the second mixed partial derivatives of $f$ are continuous on some neighbourhood of $f$, then these mixed partial derivatives are equal on this neighbourhood. That is for $\mathbf{x} = (x_1, x_2, …, x_n)$ and $z = f(x_1, x_2, …, x_n)$ we have that $f_{ij} (\mathbf{x}) = f_{ji} (\mathbf{x})$ for $i = 1, 2, …,n$ and $j = 1, 2, …, n$. So the continuity of all of the second mixed partial derivatives of $f$ imply that the Hessian $H(\mathbf{x})$ is symmetric.

## Example 1

**Find the Hessian Matrix of the function $f(x, y) = x^2y + xy^3$.**

We need to first find the first partial derivatives of $f$. We have that:

(3)

We then calculate the second partial derivatives of $f$:

(4)

Therefore the Hessian Matrix of $f$ is:

(5)

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