First of all, the 5870 and 580 use different power connections. Did you procure a nmini 6-pin to 8-pin power cable for the 580?
If so, understand that you won't get a signal during the boot process as the 580 lacks an EFI ROM. So you wouldn't see anything until your desktop (or login screen) loads. If you use FileVault then you won't see anything period, as the FileVault password entry screen is part of the EFI environment.
If you're sure you're not running FileVault and you are still getting no signal even after the machine has booted then you should try another output on the card if possible--DVI or HDMI.