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I'm trying to diagonalize the following unitary matrix:

$\frac {1}{\sqrt{5}}\begin{pmatrix} 1 &2 \\ 2i &-i \end{pmatrix}$

My approach is to find the eigenvalues and eigenvectors in the usual way. However, no matter what I do, this is not yielding me the correct eigenvalues.

By doing the usual algebra (using det(A - kI) = 0 where k is the eigenvalue), I get the following equation quadratic in k:

$k^2 + \frac {i-1}{\sqrt{5}}k - i = 0$

I then solve this quadratic equation for k using the quadratic formula with a = 1, b = $\frac {i-1}{\sqrt{5}}$ and c = -i. This gives me a pair of conjugate eigenvalues. However, they are not the correct eigenvalues!

I just wonder if my approach is incorrect. Is there a way to easily diagonalize a unitary matrix with complex entries, by using the fact that it is unitary? I know a unitary matrix will have orthogonal eigenvectors, eigenvalues of modulus 1, etc. But none of that really helps me in actually finding the eigenvalues and eigenvectors.

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  • $\begingroup$ Your approach and your characteristic equation are correct. Perhaps you solved the equation wrongly. $\endgroup$ – user1551 Nov 19 '12 at 11:56
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Your approach is correct. Here is the actual diagonalization via Wolfram Alpha.

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