Find the characteristic equation, the eigenvectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$, and the corresponding eigenvalues $\lambda_{\mathbf{a}}, \lambda_{\mathbf{b}}, \lambda_{\mathbf{c}}, \lambda_{\mathbf{d}}$ of the matrix

$$A=\left(\begin{array}{cccc} i & 1 & 0 & 0 \\ 1 & i & 0 & 0 \\ 0 & 0 & i & 1 \\ 0 & 0 & -1 & i \end{array}\right)$$

Show that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\}$ spans the complex vector space $\mathbb{C}^{4}$.

Consider the four subspaces of $\mathbb{C}^{4}$ defined parametrically by

$$\mathbf{z}=s \mathbf{a}, \quad \mathbf{z}=s \mathbf{b}, \quad \mathbf{z}=s \mathbf{c}, \quad \mathbf{z}=s \mathbf{d} \quad\left(\mathbf{z} \in \mathbb{C}^{4}, s \in \mathbb{C}\right)$$

Show that multiplication by $A$ maps three of these subspaces onto themselves, and the remaining subspace into a smaller subspace to be specified.