The magnitude of the second dominant eigenvalue of a reducible matrix, as I know, is supposed to be 1, why it's not the case for this matrix :
$$ \begin{matrix} 0 & 1 & 0 & 0 \\ 0.5 & 0 & 0.5 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{matrix} $$
This is a reducible matrix, but the eigenvalues I'm having are these :
(-0.50000, 0.50000i)
( 0.00000, 0.00000i)
( 1.00000, 0.00000i)
(-0.50000,-0.50000i)
As you can see the second dominant eigenvalue is not 1, am I missing something ?