How to obtain negative voltage from positive voltage ?
One way is to charge up a capacitor with the positive voltage, and then disconnect the charged capacitor and connect the + lead to the ground of the circuit. You'll now have a negative voltage (relative to circuit ground) at the - lead of the charge capacity. 

Of course, in practice this is typically driven by an oscillator that does the charge/disconnect/reverse cycle frequently and then uses diodes to transfer the negative charge to a holding capacitor that keep the negative voltage present continuously.