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:

${\begin{pmatrix}\nabla (f_{1})(a_{1},a_{2},\dots ,a_{n})\\\nabla (f_{2})(a_{1},a_{2},\dots ,a_{n})\\\cdot \\\cdot \\\cdot \\\nabla (f_{m})(a_{1},a_{2},\dots ,a_{n})\\\end{pmatrix}}$

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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:

${\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&{\frac {\partial f_{1}}{\partial x_{2}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&\dots &{\frac {\partial f_{1}}{\partial x_{n}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}\\{\frac {\partial f_{2}}{\partial x_{1}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&{\frac {\partial f_{2}}{\partial x_{2}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&\dots &{\frac {\partial f_{2}}{\partial x_{n}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}\\\cdot &\cdot &\cdot &\cdot \\{\frac {\partial f_{m}}{\partial x_{1}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&{\frac {\partial f_{m}}{\partial x_{2}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}&\dots &{\frac {\partial f_{m}}{\partial x_{n}}}(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}\\\end{pmatrix}}$

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.

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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:

${\begin{pmatrix}\nabla (f_{1})\\\nabla (f_{2})\\\cdot \\\cdot \\\cdot \\\nabla (f_{m})\\\end{pmatrix}}$

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:

${\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}&{\frac {\partial f_{1}}{\partial x_{2}}}&\dots &{\frac {\partial f_{1}}{\partial x_{n}}}\\{\frac {\partial f_{2}}{\partial x_{1}}}&{\frac {\partial f_{2}}{\partial x_{2}}}&\dots &{\frac {\partial f_{2}}{\partial x_{n}}}\\\cdot &\cdot &\cdot &\cdot \\{\frac {\partial f_{m}}{\partial x_{1}}}&{\frac {\partial f_{m}}{\partial x_{2}}}&\dots &{\frac {\partial f_{m}}{\partial x_{n}}}\\\end{pmatrix}}$

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.

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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.