I'm going to give you a kind of "shotgun" answers.
Check power to the monitor from the wall outlet or try a different monitor if available.
Check all connections to the monitor from the RX580. Try a different port from the video card to monitor.
Make sure the RX580 is firmly seated in its pci slot and power cable is secure going into the card and into the power supply. Try a different power cable from the power supply to the RX580..
Try a different video card in place of the RX580 if you have a spare AMD card.
When the PC boots any specific and audible "beeping" patterns that may point you in a direction. Check your motherboard documentation for trouble-shooting.
Things you didn't mention..is the RX580 a new install or have you been using it for awhile? Had you had any artifacting issues or graphical issues with the card?
Really more background info is needed for anyone to better assist. As it stands, I'm just listing the easy-fix stuff to check.