I'm trying to determine the eigenvalues for the following matrix:
$$ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -i & 0 & 0 \\ 0 & 0 & i & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix} $$
Where the first row and column share a single non zero element (same happens for last row and column). Am I understanding correctly that this reduces to finding the eigenvalues of:
$$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
and of
$$ \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{pmatrix} $$
Or am I missing some key result on block matrices? Can you provide references as to how I can simplify computations of eigenvalues for large block matrices?