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Thread: Finding minima

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    3

    Finding minima

    Hi,

    Suppose we have a $\displaystyle m\times2$ matrix $\displaystyle A$, and we have
    $\displaystyle
    B=\left[ {\begin{array}{*{20}c}
    {A_{11} - a} & {A_{12} - b} \\
    {A_{21} - a} & {A_{22} - b} \\
    {\vdots} & {\vdots} \\
    {A_{m1} - a} & {A_{m2} - b} \\
    \end{array}} \right]
    $.

    My problem is to find the values of $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle trace[(B^TB)^{-1}]$ is minimized,
    $\displaystyle
    [a,b]=\arg\min_{a,b}trace[(B^TB)^{-1}]
    $.

    Since $\displaystyle B^TB$ is a simple $\displaystyle 2\times2$ matrix, I get
    $\displaystyle trace[(B^TB)^{-1}]=\frac{f_a+f_b}{f_af_b-f_{ab}^2}
    $
    where
    $\displaystyle f_a = \sum_{i=1}^{m}(A_{i1}-a)^2
    $
    $\displaystyle
    f_b = \sum_{i=1}^{m}(A_{i2}-b)^2$
    $\displaystyle
    f_{ab} = \sum_{i=1}^{m}[(A_{i1}-a)(A_{i2}-b)]
    $

    The maximum of $\displaystyle \frac{f_a+f_b}{f_af_b-f_{ab}^2} $ is easy to get, but who can help me with finding the minimum? I have been stucked for quite a while.....

    Anyway, thanks a lot in advance,

    Creed
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  2. #2
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