Specify function given certain constraints

[noowutah]

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 $F$?

 

[gtoguy97]

From the two constraints for F, we can conclude that F is a function of the matrix elements that is zero when the two rows of the matrix are equal. This implies that F is a measure of the difference between the two rows of the matrix, and its derivative with respect to the individual matrix elements is zero when the two rows are equal. This means that F is a function that is maximized when the two rows of the matrix are different and minimized when they are equal.