I'm trying to take the continuum limit of a quantum walk, which involves writing the quantum 'coin' in exponential form. This is essentially just writing a matrix in exponential form. Most literature uses a simple $2 \times 2$ matrix at the 'coin' and then writes the matrix in the form. For example, a typical coin could be:
C = $\frac{1}{\sqrt2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $
C = $e^{-i\theta Q}$
Where $Q$ is a hermitian matrix. I understand that you can write this using Taylor expansion with $Q$ being a Pauli matrix. Now I am trying to write an $8 \times 8$ matrix in the form above, BUT I have multiple parameters (i.e. multipe $\theta$'s in the C matrix. Now, I'm unsure how to write Q if I have more than one parameter. Is $Q$ just the diagonal form of $C$? If I have multiple free parameters in my matrix $C$, then what is $\theta$?