Jacobian matrix

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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_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}

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.

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 fis 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_is 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_is 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.

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 \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.