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Math Help - Derivative wrt vector

  1. #1
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    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!
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  2. #2
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    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?
    Thanks from topsquark
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  3. #3
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    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.
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  4. #4
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    Re: Derivative wrt vector

    Quote Originally Posted by goftl View Post
    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 $c$
    Thanks from topsquark and goftl
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  5. #5
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    Re: Derivative wrt vector

    thanks this helped me a lot!
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