Jacobian Matrix

analysis linear-algebra

Definition

Jacobian Matrix

Let $D\subseteq {\mathbb{R}}^n$ be an open set, and let

$$f:D\to {\mathbb{R}}^m,f(\mathbf{x})=\left[\begin{array}{c}{f}_1(\mathbf{x})\\ f_2(\mathbf{x})\\ ⋮\\ f_m(\mathbf{x})\end{array}\right]$$

be a totally differentiable function. The Jacobian matrix of $f$ at $\mathbf{x}$ is the $m\times n$ matrix

$$J_f(\mathbf{x})=\left[\begin{array}{cccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}& \dots & \frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}& \dots & \frac{\partial f_2}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \frac{\partial f_m}{\partial x_2}& \dots & \frac{\partial f_m}{\partial x_n}\end{array}\right].$$

Its $(i,j)$-entry is

$$(J_f(\mathbf{x}){)}_{ij}=\frac{\partial f_i}{\partial x_j}(\mathbf{x}).$$

The Jacobian matrix is the matrix representation of the derivative of $f$ at $\mathbf{x}$.

Linear approximation

Near $\mathbf{x}$, the function $f$ is approximated by its first-order linear part:

$$f(\mathbf{x}+\mathbf{h})\approx f(\mathbf{x})+J_f(\mathbf{x})\mathbf{h}.$$

So the Jacobian describes how small changes in the input produce first-order changes in the output.

Dimensions

If $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, then:

$$J_f(\mathbf{x})\in {\mathbb{R}}^{m\times n}.$$

• rows correspond to output coordinates $f_1,\dots ,f_m$
• columns correspond to input variables $x_1,\dots ,x_n$

Chain rule

If

$$f:{\mathbb{R}}^n\to {\mathbb{R}}^m,g:{\mathbb{R}}^m\to {\mathbb{R}}^k,$$

then the chain rule can be written in matrix form as:

$$J_{g\circ f}(\mathbf{x})=J_g(f(\mathbf{x}))\cdot J_f(\mathbf{x}).$$

So composition corresponds to matrix multiplication of Jacobians.

Special cases

Scalar-valued function

If $f:{\mathbb{R}}^n\to \mathbb{R}$, then the Jacobian is a $1\times n$ row matrix:

$$J_f(\mathbf{x})=\left[\begin{array}{cccc}\frac{\partial f}{\partial x_1}& \frac{\partial f}{\partial x_2}& \dots & \frac{\partial f}{\partial x_n}\end{array}\right].$$

With the convention that the gradient is a column vector, this gives

$$J_f(\mathbf{x})=\nabla f(\mathbf{x}{)}^T.$$

One-dimensional input

If $f:\mathbb{R}\to {\mathbb{R}}^m$, then the Jacobian is an $m\times 1$ column matrix:

$$J_f(x)=\left[\begin{array}{c}{f}_1^{\prime }(x)\\ f_2^{\prime }(x)\\ ⋮\\ f_m^{\prime }(x)\end{array}\right].$$

Examples

Function from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$

Let

$$f(x_1,x_2)=\left[\begin{array}{c}{x}_1{x}_2\\ x_1^2+\sin x_2\end{array}\right].$$

Then

$$\frac{\partial f_1}{\partial x_1}=x_2,\frac{\partial f_1}{\partial x_2}=x_1,$$

and

$$\frac{\partial f_2}{\partial x_1}=2x_1,\frac{\partial f_2}{\partial x_2}=\cos x_2.$$

Therefore,

$$J_f(x_1,x_2)=\left[\begin{array}{cc}{x}_2& x_1\\ 2x_1& \cos x_2\end{array}\right].$$

The first row describes how the first output coordinate changes, and the second row describes how the second output coordinate changes.