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You are right, the rules to time-scale a T-years transition matrix $M_T$ are: $M_{k·T} = M_T^k$ $M_{T/k} = \sqrt[k]{M_T}$ The root of a matrix M can be obtained using the spectral decomposition: $M = P·D·P^{-1} \Longrightarrow M^k = P·D^k·P^{-1}$ where $P$ and $D$ are the eigenvectors and eigenvalue matrices of $M_T$. Note: The Perron-Frobenius tells ...