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