I was thinking about different ways a dual supply could be created, and I was wondering if something along the lines of what I made in this schematic would work. I don't have any kind of formal education in this stuff so excuse me if there is some blatent thing I'm missing.
Best Answer
Regulators won't sink (current, that is)
Your signal reference (ground) in a dual supply configuration needs to be able to both source and sink current -- in other words, it needs to be able to accept current flow into it as well as provide current flowing out of it.
However, a three-terminal regulator's single-pass-device output can only source current -- there's no path for current to flow into the regulator's OUT pin that takes the current where you want it to go (to the GND pin, in your case). Some regulators simply will not output at all (i.e. the pass device will turn off, leaving no current path from OUT to much of anywhere significant) under those circumstances, while others have a parasitic diode from OUT to IN that will turn on when you try to stuff current into OUT, causing the regulator to get confused.
Never fear though, the Rail Splitter is here!
Fortunately for you though, dual-supply signal references don't need that much current in most applications, so TI makes a part in TO-92 that is just what you are after: the TLE2426 "Rail Splitter" virtual ground IC.
This part can be used in place of your three-terminal regulator in the schematic you posted to provide a "0V" from the floating supply you want to split that can source and sink current (up to 20mA for the TLE2426 by itself, and you can also shove a power op amp voltage follower in there to buffer its output if you need more current). Furthermore, the TLE2426 output accurately tracks the half-supply mark; your proposal, even with a two-quadrant regulator, would cause asymmetric supply voltages if the 24V supply started to wander off.