Finding minima

**creed** · August 17th 2008, 05:28 AM

Hi,

Suppose we have a $m\times2$ matrix $A$ , and we have
$ 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 $a$ and $b$ such that $trace[(B^TB)^{-1}]$ is minimized,
$ [a,b]=\arg\min_{a,b}trace[(B^TB)^{-1}] $ .

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} $
where
$f_a = \sum_{i=1}^{m}(A_{i1}-a)^2 $
$ f_b = \sum_{i=1}^{m}(A_{i2}-b)^2$
$ f_{ab} = \sum_{i=1}^{m}[(A_{i1}-a)(A_{i2}-b)] $

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

**creed** · August 19th 2008, 11:10 AM

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