Let $A$ the block matrix given by the blocks: $$\tilde{A}=\begin{pmatrix} 1&-\mu&0&...&0&-\mu\\ -\mu&1&-\mu&...&0&0\\ 0&-\mu&1&...&0&0\\ ...&...&...&...&...\\ 0&0&0&...&1&-\mu\\ -\mu&0&0&...&-\mu&1 \end{pmatrix}\quad\quad A^*=\mathbf{diag}(-\mu,-\mu,...,-\mu) $$ where $\mu<1$.
So $$ A=\begin{pmatrix} \tilde{A}&A^*&0&...&0&A^*\\ A^*&\tilde{A}&A^*&0&...&0\\ 0&A^*&\tilde{A}&...&...&0\\ ...&...&...&...&...&...\\ A^*&0&...&0&A^*&\tilde{A} \end{pmatrix} $$ Am I allowed to expand in power series this matrix with something like $$A^{1/4}=\mathbb{I}+\sum_{k\geq 1} \binom{1/4}{k}(-1)^k(\mu T)^k$$ where $T=\frac{1}{\mu}(\mathbb{I}-A)$? Any references where can I study these kind of expansion?