http://galeon.hispavista.com/cinerama/actu2/matrixdarkcity.htm
(via Bluesnews.com 2003/11/12)
I think I learned that the rooftop in the opening chase scene in The Matrix was the same set as one of Dark City's-- unfortunately, I've loaned away my Matrix DVD which is where I think I heard that (one of the audio commentaries).
I don't disagree with your comparison, Kirk, but while The Matrix really grabbed you by the collars with a mega-rapid transition from "What the hell am I looking at" to "D00d, I've never seen anything like that!" Dark City has, IMO, a less philosophically polemical SF ground, one whose full depth isn't revealed until the very end as the, erm, unreality of the City is shown. Both films benefit from having strong visual direction, so the noir elements were perhaps inevitable. I'm noticing, though, that noir tends to show Los Angeles-type cities-- open downtowns where everybody drives. Matrix's "City," though Chicago-inspired and Sydney-set, lacks the concentration of a Gotham. Dark City had more claustrophia, a subway people use, as well as an inhabited sewer. :^) Very much a dark Gotham.

On to my final point: Dark City has Jennifer Connelly *and* Melissa George. hubba hubba! That'll trump Trinity anyday, IMO.