## Jacobian

In the world of linear approximations of multiple parameters and multiple outputs, the Jacobian is a matrix that tells you: if I twist this knob, how does that part of the output change?

$\large \dpi{200} \bg_white \begin{bmatrix} \partial \, F^1 \over \partial \; a & \partial \, F^1 \over \partial \; b & \partial \, F^1 \over \partial \; c & \cdots & \partial \, F^1 \over \partial \; z \\ \partial \, F^2 \over \partial \; a & \partial \, F^2 \over \partial \; b & \partial \, F^2 \over \partial \; c & \\ \\ & \vdots & & \ddots \\ \\ \partial \, F^N \over \partial \; a & \partial \, F^N \over \partial \; b & \partial \, F^N \over \partial \; c & \cdots & \partial \, F^N \over \partial \; z \end{bmatrix}$
$\dpi{200} \bg_white \large \begin{bmatrix} ^{\partial F^\mathbf{1}} \!/ _{\partial \mathbf{a}} && ^{\partial F^1} \!/ _{\partial \mathbf{b}} && ^{\partial F^1} \!/ _{\partial \mathbf{c}} && \cdots && ^{\partial F^1} \!/ _{\partial \mathbf{z}} \\ \\ ^{\partial F^\mathbf{2}} \!/ _{\partial a} && ^{\partial F^2} \!/ _{\partial b} && ^{\partial F^2} \!/ _{\partial c} && \cdots && ^{\partial F^2} \!/ _{\partial z} \\ \\ && \vdots &&&& \ddots \\ \\ ^{\partial F^\mathbf{N}} \!/ _{\partial a} && ^{\partial F^N} \!/ _{\partial b} && ^{\partial F^N} \!/ _{\partial c} && \cdots && ^{\partial F^\mathbf{N}} \!/ _{\partial \mathbf{z}} \end{bmatrix}$

(The Jacobian is defined at a point. If the space not flat, but instead only approximated by flat things that are joined together, then you would stitch together different Jacobians as you stitch together different flats.)

Pretend that a through z are parameters, or knobs you can twist. Let’s not say whether you have control over them (endogenous variables) or whether the environment / your customers / your competitors / nature / external factors have control over them (exogenous parameters).

And pretend that through F are the separate kinds of output. You can think in terms of a real number or something else, but as far as I know the outputs cannot be linked in a lattice or anything other than a matrix rectangle.

In other words this matrix is just an organised list of “how parameter c affects output F”.

Notan bene — the Jacobian is just a linear approximation. It doesn’t carry any of the info about mutual influence, connections between variables, curvature, wiggle, womp, kurtosis, cyclicity, or even interaction effects.

A Jacobian tensor would tell you how twisting knob a knocks on through parameters h, l, and p. Still linear but you could work out the outcome better in a difficult system — or figure out what happens if you twist two knobs at once.

In maths jargon: the Jacobian is a matrix filled with partial derivatives.

67 notes

1. infinitedeltas reblogged this from isomorphismes
2. guattariteenageriot reblogged this from isomorphismes and added:
No idea what this is about.
3. dataanxiety reblogged this from isomorphismes and added:
Excerpt from isomorphismes: Emphasis mine.
4. pipoytales reblogged this from isomorphismes and added:
actually understand what’s being said here? Ohnooeezzz, Pipoy...close to becoming
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9. orbitalresonance reblogged this from isomorphismes and added:
win
10. isomorphismes posted this