Jacobian

Sometimes called "Jacobian Matrix", or "differential".

Common definition

Given a function [math]f:\mathbb{R}^m\rightarrow\mathbb{R}^n[/math] (I use the convention of m first because it takes it from m to n) the:

• differential of [ilmath]f[/ilmath] at [ilmath]x[/ilmath], denoted [math]df_x[/math] or [math]Df_x[/math] which I prefer, as you often find [math]df[/math] in a fraction involving [math]dx[/math]
• Jacobian matrix of [ilmath]f[/ilmath] at [ilmath]x[/ilmath] often denoted [math]J_{f(x)}[/math]

Are given by:

[math]Df_x:\mathbb{R}^m\rightarrow\mathbb{R}^n[/math], [math]Df_x=\begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_m} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} & \cdots & \frac{\partial f_n}{\partial x_m} \end{pmatrix}[/math]

This is a n-by-m matrix using my convention.

How to remember which way round this matrix goes

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I have trouble remembering this but it is trivial to deduce. Notice [math]f:\mathbb{R}^m\rightarrow\mathbb{R}^n[/math] this means:

(m length vector)[math]\mapsto[/math](n length vector)

So for [math]x\in \mathbb{R}^m[/math] we must have [math]Df_x x\in\mathbb{R}^n[/math] (in the matrix multiplication sense)

As we go across [math]Df_x[/math] and down [math]x[/math] when multiplying, we must have columns (the width of the matrix) = length of the vector, so it is [ilmath]m[/ilmath] across, and thus [ilmath]n[/ilmath] down.

Furthermore

[math]\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \cdots \\ \vdots & \ddots & \vdots \\ \cdots & \cdots & \cdots \end{pmatrix}\times\begin{pmatrix}x\\y\\ \vdots \end{pmatrix} [/math][math]=\begin{pmatrix} \frac{\partial f_1}{\partial x}x+\frac{\partial f_1}{\partial y}y+\cdots \\ \frac{\partial f_2}{\partial x}x+\frac{\partial f_2}{\partial y}y+\cdots \\ \vdots \end{pmatrix} [/math][math]=\begin{pmatrix} \text{change in }f_1\text{per }x\text{ times }x +\text{change in }f_1\text{per }y\text{ times }y + \cdots \\ \text{change in }f_2\text{per }x\text{ times }x +\text{change in }f_2\text{per }y\text{ times }y + \cdots \\ \vdots \end{pmatrix}[/math]

Which makes perfect dimensional sense