CA systems implement all algebraic functions of square matrices, simply as the natural extension of a function $f$ with a zero at $x=0$ to diagonal matrices as arguments that map the functions to the diagonal elements by the principle of power series.
$$x\to f(x) = \sum_1^\infty \ f_n x^n$$
$$f(A) =f\ \left(\quad\left( \begin{array}{cccc}a_1&0&0&\dots\\0&a_2&0 \\0&0&a_3\\\vdots\end{array}\right)\quad\right)= \left( \begin{array}{cccc}f(a_1)&0&0&\dots\\0&f(a_2)&0\\0&0&f(a_3)\\ \vdots\end{array}\right)$$
If $f(0)\ne 0$, matrix functions become 'complex'
e.g. here for $n! =\Gamma(n+1)$
(MatrixFunction[(Gamma[# + 1] &),
IdentityMatrix[2] +
\[Alpha] PauliMatrix[1] +
\[Beta] PauliMatrix[2] +
\[Gamma] PauliMatrix[3]] //.
{\[Alpha]^2 + \[Beta]^2 + \[Gamma]^2 :> \[Phi]^2} //
FullSimplify // PowerExpand) )
$$\left(
\begin{array}{cc}
\frac{\gamma \Gamma (\phi +2)}{2 \phi }-\frac{\gamma \Gamma (2-\phi )}{2 \phi }+\frac{\Gamma (\phi +2)}{2}+\frac{\Gamma (2-\phi )}{2} & \frac{\gamma ^2 \Gamma (2-\phi )}{2 \phi (\alpha +i \beta )}-\frac{\gamma ^2 \Gamma (\phi +2)}{2 \phi (\alpha +i \beta )}+\frac{\phi \Gamma (\phi +2)}{2 (\alpha +i \beta )}-\frac{\phi \Gamma (2-\phi )}{2 (\alpha +i \beta )} \\
\frac{\alpha \Gamma (\phi +2)}{2 \phi }-\frac{\alpha \Gamma (2-\phi )}{2 \phi }+\frac{i \beta \Gamma (\phi +2)}{2 \phi }-\frac{i \beta \Gamma (2-\phi )}{2 \phi } & -\frac{\gamma \Gamma (\phi +2)}{2 \phi }+\frac{\gamma \Gamma (2-\phi )}{2 \phi }+\frac{\Gamma (\phi +2)}{2}+\frac{\Gamma (2-\phi )}{2} \\
\end{array}
\right)$$