# Finding minima

• August 17th 2008, 05:28 AM
creed
Finding minima
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
• August 19th 2008, 11:10 AM
creed
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