RFM, with no offense intended towards you or Guy, I'm quite certain your bird is not a Purplish-backed Jay for a few reasons.
1) Your bird's legs are black (PBJs' are yellow),
2) your bird's eyes are black (PBJs' are yellow),
3) your bird's bill - relative to body - is American Crow-size (not small as a PBJ's would be),
4) your bird has a glossy black-blue tone on its entire lower body - wings, tail, flanks, underside, etc. (PBJs have, first of all, not glossy, but blue/purple coloration, and this coloration is only on wings and parts of the tail (tail feathers, from available photos I could find, seem to be edged black (EDIT: my mistake - I was seeing the black undertail through ruffled tail-feathers), and flanks, underbelly, and undertail are a definite black)),
5) your bird's tail is expected length of American Crow's in relation to its body (PBJs' tails are almost half their length!), and
6) your bird is just too big. I don't have my Howell & Webb (Mexico bird book) with me right now to see measurements, but from all the photos I could find of PBJs online, they seem diminuitive and magpie-sized. Your bird is clearly a Corvus, not a Cyanocorax.
Here are the only photos online I could find of PBJs, but I think they probably speak better than my rant:
http://zettesworld.com/zooblog/030906SanBlasJay.jpg
http://www.hillcountryaviaries.com/images/purpilishbacked.jpg
http://en.wikipedia.org/wiki/Image:Purplish-backed_Jay.jpg
http://en.wikipedia.org/wiki/Image:Purplish-backed_Jay_eating.jpg
Wanted to also offer up some photos from our gallery here at BirdForum as comparatively glossy-blue-looking American Crows. Given, most aren't as glossy as your bird, but hey, that's variation for ya!
http://www.birdforum.net/pp_gallery/showphoto.php/photo/100372/sort/1/cat/all/page/1
http://www.birdforum.net/pp_gallery/showphoto.php/photo/46222/sort/1/cat/all/page/2
http://www.birdforum.net/pp_gallery/showphoto.php/photo/43446/sort/1/cat/all/page/2
http://www.birdforum.net/pp_gallery/showphoto.php/photo/77285/sort/1/cat/all/page/1
http://www.birdforum.net/pp_gallery/showphoto.php/photo/68394/sort/1/cat/all/page/2
http://www.birdforum.net/pp_gallery/showphoto.php/photo/18098/sort/1/cat/all/page/2