Hessian matrix

Definition at a point

For a function of two variables at a point

Suppose $f$ is a real-valued function of two variables $x, y$ and $(x_{0}, y_{0})$ is a point in the domain of $f$ . Suppose all the four second-order partial derivatives exist at $(x_{0}, y_{0})$ , i.e., the two pure second-order partials $f_{x x} (x_{0}, y_{0}), f_{y y} (x_{0}, y_{0})$ exist, and so do the two second-order mixed partial derivatives $f_{x y} (x_{0}, y_{0}$ and $f_{y x} (x_{0}, y_{0})$ . Then, the Hessian matrix of $f$ at $(x_{0}, y_{0})$ , denoted $H (f) (x_{0}, y_{0})$ , is a $2 \times 2$ matrix of real numbers defined as follows:

$(\begin{matrix} f_{x x} (x_{0}, y_{0}) & f_{x y} (x_{0}, y_{0}) \\ f_{y x} (x_{0}, y_{0}) & f_{y y} (x_{0}, y_{0}) \end{matrix})$

For a function of multiple variables at a point

Suppose $f$ is a real-valued function of multiple variables $(x_{1}, x_{2}, \dots, x_{n})$ . Suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ . In other words, $a_{1}, a_{2}, \dots, a_{n}$ are real numbers and the point has coordinates $x_{1} = a_{1}, x_{2} = a_{2}, \dots, x_{n} = a_{n}$ . Suppose, further, that all the second-order partials (pure and mixed) of $f$ with respect to these variables exist at the point $(a_{1}, a_{2}, \dots, a_{n})$ . Then, the Hessian matrix of $f$ at $(a_{1}, a_{2}, \dots, a_{n})$ , denoted $H (f) (a_{1}, a_{2}, \dots, a_{n})$ , is a $n \times n$ matrix of real numbers defined as follows:

The $(i j)^{t h}$ entry (i.e., the entry in the $i^{t h}$ row and $j^{t h}$ column) is $f_{x_{i} x_{j}} (a_{1}, a_{2}, \dots, a_{n})$ . This is the same as $\frac{\partial^{2} f}{\partial x_{j} \partial x_{i}} f (a_{1}, a_{2}, \dots, a_{n})$ . Note that in the two notations, the order in which we write the partials differs because the convention differs (left-to-right versus right-to-left).

The matrix looks like this:

Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \begin{pmatrix} f_{x_1x_1}(a_1,a_2,\dots,a_n) & f_{x_1x_2}(a_1,a_2,\dots,a_n) & \dots & f_{x_2x_n}(a_1,a_2,\dots,a_n)\\ f_{x_2x_1}(a_1,a_2,\dots,a_n) & f_{x_2x_2}(a_1,a_2,\dots,a_n) & \dots & f_{x_2x_n}(a_1,a_2,\dots,a_n)\\ \dot & \dot & \dot & \dot\\ \dot & \dot & \dot & \dot\\ \dot & \dot & \dot & \dot\\ f_{x_nx_1}(a_1,a_2,\dots,a_n) & f_{x_nx_2}(a_1,a_2,\dots,a_n) & \dots & f_{x_nx_n}(a_1,a_2,\dots,a_n)\\\end{pmatrix}}

Definition as a function

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x, y$ . The Hessian matrix of $f$ , denoted $H (f)$ , is a $2 \times 2$ matrix-valued function that sends each point to the Hessian matrix at that point, if that matrix is defined. It is defined as:

$(x_{0}, y_{0}) \mapsto H (f) (x_{0}, y_{0}) = (\begin{matrix} f_{x x} (x_{0}, y_{0}) & f_{x y} (x_{0}, y_{0}) \\ f_{y x} (x_{0}, y_{0}) & f_{y y} (x_{0}, y_{0}) \end{matrix})$

In the point-free notation, we can write this as:

$H (f) = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix})$

Under continuity assumptions

If we assume that all the second-order partials of $f$ are continuous functions everywhere, then the following happens:

The Hessian matrix of $f$ at any point is a symmetric matrix, i.e., its $(i j)^{t h}$ entry equals its $(j i)^{t h}$ entry. This follows from Clairaut's theorem on equality of mixed partials.
$f$ is twice differentiable as a function. Hence, the Hessian matrix of $f$ is the same as the Jacobian matrix of the gradient vector $\nabla f$ , where the latter is viewed as a vector-valued function.

Note that the second conclusion actually only requires the existence of the gradient vector, hence it holds even if the second-order partials are not continuous.