MY APOLOGIES!
The question specifies that A is a linear map on a TWO-DIMENSIONAL complex Euclidean space. I FAIL.
I would like some direction on this problem please:
Let A be a complex n x n matrix, with ||A||^{2 }= ||A^{2}||. Prove that A is normal.
I believe the norm being used is the norm which takes the largest value of |Ax| over all vectors x of length 1. Unfortunately, and quite annoyingly, the problem set doesn't specify. However, there was another problem which used an unspecified norm, and this norm seemed to be the one which made sense, so I believe this norm is the one being used.
Here's my idea: first assume that A is upper-triangular. Prove that A must be diagonal. Then use Schur's theorem (a matrix can be unitarily upper triangularized) to prove the general case. But the reduction to upper triangular matrices hasn't yet helped me very much.
Another idea is to show that x*AA*x = x*A*Ax for all x, from which the normality of A follows. A possibly-relevant fact is that ||A|| is the largest singular value of A. Perhaps I can use the factorization for some unitary U. Obviously, none of these have yet led me to a solution. Any ideas?