# Derivative wrt vector

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• Aug 13th 2014, 04:56 AM
goftl
Derivative wrt vector
I need to minimize
$\frac{1}{||x||^2}||ax-\mu x||^2$,
where x, a and $\mu$ are vectors in $R^n$.
My Idea was to take the derivative, set it to zero and solve for x, but I'm not good in matrix calculus. I would be extremely thankful if somebody could help me here!
• Aug 13th 2014, 05:23 AM
HallsofIvy
Re: Derivative wrt vector
If $a$, $\mu$, and $x$ are vectors, how are you defining " $ax$" and " $\mu x$"? Is that the dot product? But then it wouldn't make sense to take the norm of $ax- \mu x$ which would be a number! Are you sure that $a$ and $\mu$ are not scalars?
• Aug 13th 2014, 05:51 AM
goftl
Re: Derivative wrt vector
yes, the dot product is meant here and the result of the term should be a number. I need to find the optimal value of x given a and mu.
• Aug 13th 2014, 07:59 AM
romsek
Re: Derivative wrt vector
Quote:

Originally Posted by goftl
yes, the dot product is meant here and the result of the term should be a number. I need to find the optimal value of x given a and mu.

you can just combine a and $\mu$ into

$c=a-\mu$ and then find $x$ to minimize

$\dfrac {\|c\cdot x\|^2}{\|x\|^2}$

This is clearly minimized if $x$ is orthogonal to $c$

As $c\in \mathbb{R}^n$ there is an (n-1) dimensional subspace containing all these vectors orthogonal to $c$
• Aug 15th 2014, 02:50 AM
goftl
Re: Derivative wrt vector
thanks this helped me a lot!