Absolutely. ATA is the "specification" and EIDE is an acronym about a Enhanced IDE or "Integrated Drive Electronics." As such, one thing is vaguely related to the other and creates some confusion until you realize that IDE is all you need to know about and that IDE cables come in TWO FLAVORS.
80 Conductor and 40 Conductor. Both are usable in 40 Pin IDE connections, but for best effect I just get all 80 conductor cables, set the drives to Cable Select and install.
More definitions at http://www.google.com/search?hl=en&lr=&ie=UTF-8&oi=defmore&q=define:EIDE
What is the difference between ATA and EIDE connections?
Are the two interchangeable?