Can I output a HDMI signal with an Arduino?

No, not directly. Arduinos just don't have the horsepower to do such a task. For this project, I would recommend using a Raspberry Pi. Take a look at this awesome blog post by Joonas Pihlajamaa on using a Raspberry Pi as a Arduino HDMI shield.

Please see Chrontel's products at www.chrontel.com:

• CH7035 - TTL to HDMI output.
• CH7026 - TTL to CVBS.
• CH7033 - TTL to VGA and HDMI.
• CH7034 - TTL to VGA.
• CH7322 - HDMI CEC.

Their TTL input supports RGB 8-8-8. 5-6-5, YCrCb 4:2:2, ITU656, etc. I think the Arduino can use their MCU interface to write graphics data to their frame buffer directly. On-chip scaler can scale frame buffer content to all HDMI output resolutions like 1080P.