## help for finding the 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