TITLE: complexity of eigenvalue decomposition
QUESTION [20 upvotes]: what is the computational complexity of eigenvalue decomposition for a unitary matrix?
is O(n^3) a correct answer?

REPLY [2 votes]: I think the other answers are wrong. I periodically look up this problem and I believe it to be open. I will summarize my opinion:

The symmetric eigenvalue problems is "solved". Wilkinson was able to prove that the QR iteration, with his own special shift strategy, converges cubically. See this for example.
The nonsymmetric eigenvalue problem is still open. The chapter on that subject in Golub and Van Loan says has a discussion on how the Wilkinson shift fails on some nonsymmetric matrices. They also mention that Wilkinson's ad-hoc shift should not be taken "too seriously" and that really it only gives the QR iteration a fresh start and a chance at better convergence.

As far as I can tell, nobody knows the computational complexity of the approximate eigenvalue problem.
Edit: I stand corrected. Thanks Suvrit.