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