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

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 $f_1(x_1,x_2,\dots ,x_n),f_2(x_1,x_2,\dots ,x_n),\dots ,f_m(x_1,x_2,\dots ,x_n)$. 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 matrix of $f$ at $(a_1,a_2,\dots ,a_n)$ is a $m\times n$ matrix of numbers whose $i^{th}$ row is given by the gradient vector of $f_i$ at $(a_1,a_2,\dots ,a_n)$.

Explicitly, in terms of rows, it looks like:

$\left(\begin{array}{c}\mathrm{\nabla }(f_1)(a_1,a_2,\dots ,a_n)\\ \mathrm{\nabla }(f_2)(a_1,a_2,\dots ,a_n)\\ \cdot \\ \cdot \\ \cdot \\ \mathrm{\nabla }(f_m)(a_1,a_2,\dots ,a_n)\\ \end{array}\right)$

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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 $f_1(x_1,x_2,\dots ,x_n),f_2(x_1,x_2,\dots ,x_n),\dots ,f_m(x_1,x_2,\dots ,x_n)$. 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 matrix of $f$ at $(a_1,a_2,\dots ,a_n)$ is a $m\times n$ matrix of numbers whose $(ij{)}^{th}$ entry is given by:

$\frac{\partial f_i}{\partial x_j}(x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}$

Here's how the matrix looks:

$\left(\begin{array}{cccc}(x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& \dots & (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}\\ (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& \dots & (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}\\ \cdot & \cdot & \cdot & \cdot \\ (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}& \dots & (x_1,x_2,\dots ,x_n){|}_{(x_1,x_2,\dots ,x_n)=(a_1,a_2,\dots ,a_n)}\\ \end{array}\right)$

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 $f_1(x_1,x_2,\dots ,x_n),f_2(x_1,x_2,\dots ,x_n),\dots ,f_m(x_1,x_2,\dots ,x_n)$. Then, the Jacobian matrix of $f$is a $m\times n$ matrix of functions whose $i^{th}$ row is given by the gradient vector of $f_i$. Explicitly, it looks like this:

$\left(\begin{array}{c}\mathrm{\nabla }(f_1)\\ \mathrm{\nabla }(f_2)\\ \cdot \\ \cdot \\ \cdot \\ \mathrm{\nabla }(f_m)\\ \end{array}\right)$

Note that the domain of this function is the set of points at which all the $f_i$s individually are differentiable.

Definition 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 $f_1(x_1,x_2,\dots ,x_n),f_2(x_1,x_2,\dots ,x_n),\dots ,f_m(x_1,x_2,\dots ,x_n)$. Then, the Jacobian matrix of $f$ is a $m\times n$ matrix of functions whose $(ij{)}^{th}$ 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. Here's how the matrix looks:

$\left(\begin{array}{cccc}& & \dots & \\ & & \dots & \\ \cdot & \cdot & \cdot & \cdot \\ & & \dots & \\ \end{array}\right)$

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.

Particular cases

Case	What happens in that case?
$m=n=1$	$f$ is a real-valued function of one variable. The Jacobian matrix is a $1\times 1$ matrix whose entry is the ordinary derivative.
$n=1$, $m>1$	$f$ is a vector-valued function of one variable. We can think of it as a parametric curve in ${\mathbb{R}}^m$. The Jacobian matrix is a $m\times 1$ matrix which, read as a column vector, is the parametric derivative of the vector-valued function.
$m=1$, $n>1$	$f$ is a real-valued function of multiple variables. The Jacobian matrix is a $1\times n$ matrix which, read as a row vector, is the gradient vector function.
$f$ is a linear or affine map.	The Jacobian matrix is the same as the matrix describing $f$ (or, if $f$ is affine, the matrix describing the linear part of $f$).
$m=n$, and we are identifying the spaces of inputs and outputs of $f$.	The Jacobian matrix can then be thought of as a linear self-map from the $n$-dimensional space to itself. In this context, we can consider the Jacobian determinant.