The tv screen has now gone black.
There was some, probably important, information I left out. I specifically wanted to know if anyone knew why the tv was being detected by the graphics card only when the tv was on a different hdmi input, and I did not want to distract from that.
Here is the other relevant information. I found the tv on the side of the road with a hand written sign that said "Take it! Has power, but no signal". The last thing I need is more junk in my life, but I thought to myself that if it was capable of displaying that it had no signal that it might work on some inputs even if some don't work. If I was lucky, the only problem with the tv would be user error.
So I loaded it up and took it home. Everything seemed to work when I plugged it in. I did not have an hdmi cable, though, so I ordered a 10' hdmi cable from Amazon for $4 and waited for it to arrive. After the cable arrived I plugged it into the first hdmi slot and received no signal. I checked the other hdmi slots and they all seemed to work. However, for some reason I could not get hdmi audio to output from the tv even though everything appeared to be working correctly in the computer. Also, nvidia reported that the display did not support hdcp, thought I know it does.
Everything worked fine like this for a few weeks. Then yesterday I decided to check if swapping the ports would make audio or hdcp start working. For a moment I plugged the cable into the side hdmi port which kind of crimped the cable at an odd angle. I was worried that I might have damaged the cable but it appeared to still work fine. Later on, though, the display suddenly lost signal from the computer. I tried swapping ports and at first the screen would display and then switch off. Then it started to exhibit the symptoms I mentioned in the original post. Today I decided to test the component plugs and the analog speaker output that I had never tested. I did this and they worked. I left everything plugged in and went about the business of the day.
This evening I borrowed a bluray player and a high quality hdmi cable to try to confirm whether it was the tv or the cable or the computer. I switched on the tv to test it and the screen stayed black. The power light turns from orange to white but I cannot see any sign of light from inside the tv. The sound works through tv and through component/av, but I could not get the sound to work through hdmi. I unplugged everything from the tv including the power and let it sit for a while. When I turn it on the screen still stays black.