The 'classic' non-inverting amplifier configuration uses negative feedback. What happens when the + and - terminals are switched? Based on my pencil-and-paper analysis and a Circuitlab simulation, it seems that the result is the same. Is this correct? Can we make a non-inverting amplifier with positive feedback?

Derivation for the positive feedback non-inverting amplifier: I assume that input impedance is infinite, output impedance is zero, and the output voltage is given by 

$$ V\_{out} = A(V^{+} - V^{-}) $$

Now using the voltage divider equation, 

$$ V^{+} = V\_{out} \frac{R\_b}{R\_a+R\_b} $$

and for the negative terminal, we have

$$ V^{-} = V\_{in} $$ 

Substituting into the equation for the output voltage,

$$ V\_{out} = A(V^{+} - V^{-}) = AV\_{out} \frac{R\_b}{R\_a + R\_b} - AV\_{in} $$

$$ V\_{out}(1 - A \frac{R\_b}{R\_a + R\_b}) = -AV\_{in} $$

$$ V\_{out} = \frac{-AV\_{in}}{1 - A \frac{R\_b}{R\_a + R\_b}}$$

If the input frequency is relatively low, and both resistors are of a similar order of magnitude, then 

$$ A \frac{R\_b}{R\_a + R\_b} >> 1$$

$$ V\_{out} \approx \frac{-AV\_{in}}{ - A \frac{R\_b}{R\_a + R\_b}} = 
V\_{in} \frac{Ra + R\_b}{R\_b} $$

Which is the formula commonly given for the output of the classic non-inverting amplifier. So is there any difference between these two circuits? If so, where does my analysis fail? What is a more accurate way of analyzing the positive feedback circuit presented below? 

[![Non-inverting Amplifiers](https://i.stack.imgur.com/BirIV.png)](https://i.stack.imgur.com/BirIV.png)

EDIT: As suggested by Jasen, the positive feedback circuit I posted is a Schmitt Trigger. The Wikipedia article on [Schmitt Triggers](https://en.wikipedia.org/wiki/Schmitt_trigger#Op-amp_implementations) explains it fairly well. It turns out that they key to analyzing it is to solve for the voltages at the input terminals and then treat the Op-Amp as a comparator. That is, find equations for V+ and V-, and then ask the question "When does V+ > V- hold?"