Once you complete all the Archives in Normal difficulty, Hard mode will unlock (as will a bonus Archive, the Hope Archive, which is a 10 stage, linked to other achievements).
Back in Lumi's Garden (where you select which Archive to attack next), go into the menu, and select Difficulty to toggle Normal to Hard.
Then you can attempt each Archive in Hard mode.
The Matrix Archive wasn't really much harder than normal mode. I finished it first time. If you're good enough to finish the game on Normal, this should be a cakewalk.