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what is the computational complexity of eigenvalue decomposition for a unitary matrix? is O(n^3) a correct answer?

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I have some doubts about the relevance of the answers given below. You cannot compute the eigenvalues of a general unitary matrix in finite time. Because, this calculations could be used to solve every polynomial equation with real roots (the real axis is transformed rationally into the unit circle). –  Denis Serre Apr 25 '11 at 20:03
Add "...up to a required approximation" to solve this issue. Otherwise you are right, one cannot compute them in any finite time with the usual set of operations. –  Federico Poloni Jan 21 '14 at 22:44

3 Answers 3

In practice, $O(n^3)$.

In theory, it has the same complexity of matrix multiplication and more or less all the "in practice $O(n^3)$" linear algebra problems, that is, $O(n^\omega)$ for some $2<\omega<2.376$. For this last assertion, see Demmel, Dimitriu, Holtz, "Fast linear algebra is stable".

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Take a look at the following link (and references therein) for the complexity of various algorithms for common mathematical operations:

Computational Complexity of Mathematical Operations.

In particular, the complexity of the eigenvalue decomposition for a unitary matrix is, as it was mentioned before, the complexity of matrix multiplication which is $O(n^{2.376})$ using the Coppersmith and Winograd algorithm.

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The current content of that link does not describe the complexity of eigenvalue decomposition, although its content is very good. –  zhanxw Dec 6 '13 at 18:38

Yep O(n^3) is right

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