Jacobian matrix

This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative

Importance

The Jacobian matrix is the appropriate notion of derivative for a function that has multiple inputs (or equivalently, vector-valued inputs) and multiple outputs (or equivalently, vector-valued outputs).

Definition at a point

Direct epsilon-delta definition

Definition at a point in terms of gradient vectors as row vectors

Suppose $f$ is a vector-valued function with $n$ -dimensional inputs and $m$ -dimensional outputs. Explicitly, suppose $f$ is a function with inputs $x_{1}, x_{2}, \dots, x_{n}$ and outputs $f_{1} (x_{1}, x_{2}, \dots, x_{n}), f_{2} (x_{1}, x_{2}, \dots, x_{n}), \dots, f_{n} (x_{1}, x_{2}, \dots, x_{n})$ . Suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ such that $f_{i}$ is differentiable at $(a_{1}, a_{2}, \dots, a_{n})$ for $i \in {1, 2, \dots, m}$ . Then, the Jacobian matrixof $f$ at $(a_{1}, a_{2}, \dots, a_{n})$ is a $m \times n$ matrix of numbers whose $i^{t h}$ row is given by the gradient vector of $f_{i}$ at $(a_{1}, a_{2}, \dots, a_{n})$ .

Definition at a point in terms of partial derivatives

Suppose $f$ is a vector-valued function with $n$ -dimensional inputs and $m$ -dimensional outputs. Explicitly, suppose $f$ is a function with inputs $x_{1}, x_{2}, \dots, x_{n}$ and outputs $f_{1} (x_{1}, x_{2}, \dots, x_{n}), f_{2} (x_{1}, x_{2}, \dots, x_{n}), \dots, f_{n} (x_{1}, x_{2}, \dots, x_{n})$ . Suppose $(a_{1}, a_{2}, \dots, a_{n})$ is a point in the domain of $f$ such that $f_{i}$ is differentiable at $(a_{1}, a_{2}, \dots, a_{n})$ for $i \in {1, 2, \dots, m}$ . Then, the Jacobian matrixof $f$ at $(a_{1}, a_{2}, \dots, a_{n})$ is a $m \times n$ matrix of numbers whose $(i j)^{t h}$ entry is given by:

$\frac{\partial f_{i}}{\partial x_{j}} (x_{1}, x_{2}, \dots, x_{n}) |_{(x_{1}, x_{2}, \dots, x_{n}) = (a_{1}, a_{2}, \dots, a_{n})}$

Note that for this definition to be correct, it is still necessary that the gradient vectors exist. If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.

Definition as a function

Definition in terms of gradient vectors as row vectors

Suppose $f$ is a vector-valued function with $n$ -dimensional inputs and $m$ -dimensional outputs. Explicitly, suppose $f$ is a function with inputs $x_{1}, x_{2}, \dots, x_{n}$ and outputs $f_{1} (x_{1}, x_{2}, \dots, x_{n}), f_{2} (x_{1}, x_{2}, \dots, x_{n}), \dots, f_{n} (x_{1}, x_{2}, \dots, x_{n})$ . Then, the Jacobian matrixof $f$ is a $m \times n$ matrix of functions whose $i^{t h}$ row is given by the gradient vector of $f_{i}$ .

Note that the domain of this function is the set of points at which all the $f_{i}$ s individually are differentiable.

Definition at a point in terms of partial derivatives

Suppose $f$ is a vector-valued function with $n$ -dimensional inputs and $m$ -dimensional outputs. Explicitly, suppose $f$ is a function with inputs $x_{1}, x_{2}, \dots, x_{n}$ and outputs $f_{1} (x_{1}, x_{2}, \dots, x_{n}), f_{2} (x_{1}, x_{2}, \dots, x_{n}), \dots, f_{n} (x_{1}, x_{2}, \dots, x_{n})$ . Then, the Jacobian matrixof $f$ is a $m \times n$ matrix of numbers whose $(i j)^{t h}$ entry is given by:

$\frac{\partial f_{i}}{\partial x_{j}} (x_{1}, x_{2}, \dots, x_{n})$

wherever all the $f_{i}$ s individually are differentiable in the sense of the gradient vectors existing. If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.