First thing I would do is to unplug the TV and leave it unplugged a few minutes. It is uncommon but sometimes a TV can lock up just like any piece of electronics that has a micro to control it. Also make sure the electrical outlet is working properly by plugging in another device you know is functioning. If we determine it is in fact the TV then there are a few things that could cause this. I would have a hard time giving a detailed answer without knowing the model number and the exact symptom.
The problem is that RCA makes CRT rear projection TVs but has also sold DLP rear projection models. The more common is the CRT rear projection and I have seen a lot of their ITC222 chassis. If you can confirm that this is your chassis (they label it on the back with the model and serial) then I would first check for some bulged caps on the power supply. A technician should be able to visually identify capacitors that have bulged up on their heads and creating a power supply problem which is what you are describing. If not then the HV circuit could be loading down the supply. This could be the flyback, the HV splitter, or even the CRTs.
If it is a DLP then I would still suspect the power supply since lamp errors or engine failures tend to not cycle the power on and off. Unfortunately there's no way to tell if these are your problems or some other small component that has failed. I would shop around for a diagnostic on your TV because it might be worth repairing. A 55" screen is still a hefty price tag even with the new technology.