# 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

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?