Vector Differentiation

Just a quick cheatsheet on derivatives (of scalars and vectors) wrt of a vector. This is borrowed from the wiki page : Matrix Calculus.

Vector Calculus CAS Tricks

The following document is borrowed from here.

Oftentimes, we need to get linear approximation of a complicated function. This is ho you can get one with maxima

f(x,y) := exp(x^2) * sin(y);
taylor(f(x,y), [x,y], [1,2], 1);

Maxima Resources

Notations

Usually, in print following notations are in use:

A : Matrix (capital and bold)
b : Vector (small and bold)
c : scalar (small and not bold)

The rules for derivatives of a scalar by a vector

Screenshot from 2018-04-24 10:44:57.png
Screenshot from 2018-04-24 10:46:31.png

The rules for derivatives of a vector by a vector

Screenshot from 2018-04-24 10:12:26.png

Using all these rules, I have derived the commonly used least squares equation. The error function is a scalar. The optimization variable (the unknown is a vector). We are trying to minimize the error, hence we need to take the gradient (vector derivative) of error with respect to the optimization variable( the unknown) and set it to zero-vector. The values that make the gradient zeros will also be optimal value for the error (here minimum).

IMG_20180424_103301.jpg

Hope this helps!

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