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Math Help - Finding minima

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    3

    Finding minima

    Hi,

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

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

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

    The maximum of \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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