Basic tools
For $2 \times 2$ matrices the characteristic polynomial is: $ p(\lambda) = \lambda^{2} - \lambda\, \text{tr }\mathbf{A} + \det \mathbf{A} $
The roots of this function are the eigenvalues, $\lambda_{k}$, k=1,2$.
The eigenvectors solve the eigenvalue equation $ \mathbf{A} u_{k} = \lambda_{k} u_{k} $
Case 1 $ \mathbf{A} = % \left( \begin{array}{cc} i & 1 \\ 0 & -1+i \\ \end{array} \right) $ The trace and determinant are $ \text{tr } \mathbf{A} = -1 + 2 i, \qquad \det \mathbf{A} = -1 - i $ The characteristic polynomial is $ p(\lambda) = \lambda ^2+(1-2 i) \lambda +(-1-i) $ The roots of this polynomial are the eigenvalues: $ \lambda \left( \mathbf{A} \right) = \left\{ i, -1 + i \right\} $
First eigenvector: $ \begin{align} \left( \mathbf{A} - \lambda_{1} \mathbf{I}_{2} \right) u_{1} &= \mathbf{0} \\[3pt] % \left( \begin{array}{cc} i & 1 \\ 0 & -1+i \\ \end{array} \right) - \left( -1 + i \right) % \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{cc} u_{x} \\ u_{y} \\ \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \\ % \left( \begin{array}{cc} u_{x} + u_{y}\\ 0 \\ \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) % \end{align} $ The result is $ u_{1} = \left( \begin{array}{cc} u_{x} \\ u_{y} \\ \end{array} \right) = \alpha \left( \begin{array}{r} 1 \\ -1 \\ \end{array} \right), \quad \alpha \in \mathbb{C} $
Second eigenvector: $ \begin{align} \left( \mathbf{A} - \lambda_{2} \mathbf{I}_{2} \right) u_{2} &= \mathbf{0} \\[3pt] % \left( \begin{array}{cr} 0 & 1 \\ 0 & -1 \\ \end{array} \right) % \left( \begin{array}{cc} u_{x} \\ u_{y} \\ \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \\ % \end{align} $ The result is $ u_{2} = \alpha \left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right), \quad \alpha \in \mathbb{C} $
Case 2 $ \mathbf{A} = % \left( \begin{array}{cr} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{array} \right) $ The trace and determinant are $ \text{tr } \mathbf{A} = 2 \cos \theta, \qquad \det \mathbf{A} = \cos^{2} \theta + \sin^{\theta} = 1 $ The characteristic polynomial is $ p(\lambda) = \lambda ^2+(1-2 i) \lambda +(-1-i) $ The roots of this polynomial are the eigenvalues: $ \lambda \left( \mathbf{A} \right) = \left\{ \cos \theta -i \sin \theta ,\cos \theta +i \sin \theta \right\} $
First eigenvector: $ \begin{align} \left( \mathbf{A} - \lambda_{1} \mathbf{I}_{2} \right) u_{1} &= \mathbf{0} \\[3pt] % \left( \begin{array}{cr} i \sin \theta & -\sin \theta \\ \sin \theta & i \sin \theta \\ \end{array} \right) % \left( \begin{array}{cc} u_{x} \\ u_{y} \\ \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \\ % \left( \begin{array}{cc} -u_{y} \sin \theta +i u_{x} \sin \theta \\ u_{x} \sin \theta +i u_{y} \sin \theta \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) % \end{align} $ The result is $ u_{1} = \left( \begin{array}{cc} u_{x} \\ u_{y} \\ \end{array} \right) = \alpha \left( \begin{array}{r} i \\ -1 \\ \end{array} \right), \quad \alpha \in \mathbb{C} $
Second eigenvector: $ \begin{align} \left( \mathbf{A} - \lambda_{2} \mathbf{I}_{2} \right) u_{2} &= \mathbf{0} \\[3pt] % \left( \begin{array}{rr} -i \sin \theta & -\sin \theta \\ \sin \theta & -i \sin \theta \\ \end{array} \right) % \left( \begin{array}{c} u_{x} \\ u_{y} \\ \end{array} \right) &= \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \\ % \end{align} $ The result is $ u_{2} = \alpha \left( \begin{array}{c} i \\ 1 \\ \end{array} \right), \quad \alpha \in \mathbb{C} $