Thread:

1. 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

2. up...
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