State the Fundamental Theorem of Algebra. Define the characteristic equation for an arbitrary $3 \times 3$ matrix $A$ whose entries are complex numbers. Explain why the matrix must have three eigenvalues, not necessarily distinct.

Find the characteristic equation of the matrix

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

and hence find the three eigenvalues of $A$. Find a set of linearly independent eigenvectors, specifying which eigenvector belongs to which eigenvalue.