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.

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

Your approach is correct. Here is the actual diagonalization via Wolfram Alpha.


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