1.2 Jacobians

We define $\mathbf{J}$ to be the negative Jacobian (differential) of the error $\mathbf{v}$ as a function of $\mathbf{x}$ :

$\displaystyle \mathbf{J}\equiv-\dfrac{\partial\mathbf{v}(\mathbf{x})}{\partial\mathbf{x}}$

(10)

We use the negative Jacobian because when $\mathbf{v}\equiv\mathbf{z}\ominus\mathbf{h}(\mathbf{x})$ , it is more natural to compute the Jacobian of $\mathbf{h}(\mathbf{x})$ , and then

$\displaystyle \mathbf{J}=\dfrac{\partial\mathbf{h}(\mathbf{x})}{\partial\mathbf{x}}$

(11)

As with $\mathbf{v}$ and $\mathbf{R}$ , for independent errors the whole Jacobian is just the stacked matrix of individual Jacobians:

$\displaystyle \mathbf{J}_{i}$	$\displaystyle \equiv$	$\displaystyle -\dfrac{\partial\mathbf{v}_{i}(\mathbf{x})}{\partial\mathbf{x}}$
$\displaystyle \mathbf{J}$	$\displaystyle =$	$\displaystyle \left(\begin{array}{c} \mathbf{J}_{1}\\ \vdots\\ \mathbf{J}_{M} \end{array}\right)$

Ethan Eade 2012-02-16