I am trying to find the system of eigenvalues and corresponding normalized eigenvectors for the following Hermitian matrix:
$$\mathbf{H}=\begin{pmatrix}10 & 3i \\ -3i & 2\end{pmatrix}$$
I began by solving the equation $\det(\mathbf{H}-\lambda\mathbf{I}_{2\times2})$, giving:
$$\det(\mathbf{H}-\lambda\mathbf{I}_{2\times2})=\lambda^2-12\lambda+11 \implies \lambda_{1}=1,\, \lambda_{2}=11$$
We then find our corresponding eigenvectors as follows:
$$\begin{pmatrix}10 & 3i \\ -3i & 2\end{pmatrix}\begin{pmatrix}x_{1} \\ y_{1}\end{pmatrix}=\begin{pmatrix}x_{1} \\ y_{1}\end{pmatrix} \implies x_1=k,\, y_1=3ik \\ \begin{pmatrix}10 & 3i \\ -3i & 2\end{pmatrix}\begin{pmatrix}x_{2} \\ y_{2}\end{pmatrix}=11\begin{pmatrix}x_2 \\ y_2\end{pmatrix} \implies x_2 = k,\, y_2=-\frac{i k}{3}$$
Normalizing this we get:
$$\vec{v}^{1}=\frac{1}{\sqrt{10}}\begin{pmatrix}1 \\ 3i\end{pmatrix} \implies \langle\vec{v}^{1},\vec{v}^{1}\rangle = 1\\ \vec{v}^{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ -i\end{pmatrix} \implies \langle\vec{v}^{2},\vec{v}^{2}\rangle=1$$
However, this would imply that $\mathbf{U}^{\dagger}\mathbf{H}\mathbf{U}=\mathbf{\Lambda}$, with:
$$\mathbf{U}=\begin{pmatrix}\frac{1}{\sqrt{10}} & \frac{1}{\sqrt{2}} \\ \frac{3i}{\sqrt{10}} & -\frac{i}{\sqrt{2}}\end{pmatrix} \text{ and } \mathbf{\Lambda}=\begin{pmatrix}1 & 0 \\ 0 & 11\end{pmatrix}$$
However, when actually computing $\mathbf{U}^{\dagger}\mathbf{H}\mathbf{U}$ I get something very different to $\mathbf{\Lambda}$ (it's not even diagonal). So I have clearly done something very wrong, but I'm unsure where the error lies?
Looks to me like $v_1=(3i,1)\implies \|v_1\|=\sqrt{10}$ so $\displaystyle{v_1\over \|v_1\|}={1\over \sqrt{10}}(3i,1)$.
And $v_2=(-i,3)\implies \|v_2\|=\sqrt{10}$ so $\displaystyle{v_2\over \|v_2\|}={1\over \sqrt{10}}(-i,3)$.
Then $$U=\begin{bmatrix} v_1 & v_2\end{bmatrix}={1\over \sqrt{10}}\begin{bmatrix} 3i & -i\\ 1 & 3\end{bmatrix}$$ and $$U^*HU=\begin{bmatrix} 11 & 0\\ 0 & 1\end{bmatrix}.$$