Buzzing/Grinding Noise after starting

[CMat]

I have a 2004 Aztek and love it excpet there is one problem. After I start it, it makes this grinding noise about 30 seconds to one minute after being started. It never makes the noise again until I go to start it again. If I start it immediately after it was previously started, then most likely, it doesn't do it. But after it sits for about 5 minutes it will do it again. The noise only lasts for 5-10 seconds, but its rather loud...any ideas?
Thanks!

[MagicRat]

It might be an engine-driven accessory, like the power steering pump, alternator or water pump. (I would suspect the power steering pump first. Check its fluid level)

Try removing the accessory belt ("fanbelt") from the engine, then try starting the engine for about 10 seconds when cold.
If the noise disappears, then you know its an accessory. You can put the belt back, then start it and listen carefully to see if you can pinpoint the noise.

[The Silver Goat]

Could be the starter is not disengaging competely once the enginne fires...