# Specify function given certain constraints

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1. Dec 16, 2015

### stlukits

Let $F:V\rightarrow{}\mathbb{R}^{+}_{0}$ be a differentiable function. $V$ is the set of all positive real-valued $2\times{}2$ matrices, so

$$V=\left\{\left[ \begin{array}{cc} a & b \\ c & d \\ \end{array}\right]\mbox{ with }a,b,c,d\in\mathbb{R}^{+}\right\}$$

Here are the two constraints for $F$:

(1) $F\left(\left[ \begin{array}{cc} a & b \\ c & d \\ \end{array}\right]\right)=0$ if and only if $\left( \begin{array}{cc} a & b \\ \end{array}\right)=\left( \begin{array}{cc} c & d \\ \end{array}\right)$

(2) and the following:

$$\begin{array}{rlc} \displaystyle\frac{\partial{}F}{\partial{}a{}}\left(\left[\begin{array}{cc} \frac{b{}-1}{d{}-1}c{} & b \\ c & d \\ \end{array}\right] \right)&=&0 \\ \displaystyle\frac{\partial{}F}{\partial{}b{}}\left(\left[\begin{array}{cc} a & \frac{a{}-1}{c{}-1}d{} \\ c & d \\ \end{array}\right] \right)&=&0 \\ \displaystyle\frac{\partial{}F}{\partial{}c{}}\left(\left[\begin{array}{cc} a & b \\ \frac{d{}-1}{b{}-1}a{} & d \\ \end{array}\right] \right)&=&0 \\ \displaystyle\frac{\partial{}F}{\partial{}d{}}\left(\left[\begin{array}{cc} a & b \\ c & \frac{c{}-1}{a{}-1}b{} \\ \end{array}\right] \right)&=&0 \\ \end{array}$$

What can I tell about $F$?

2. Dec 21, 2015