I am working on a ratings transition matrix and I wondered how people scale it down to shorter time periods (although one should more or less stick to the estimation period i know).
It is clear that one can scale it to a longer time period by repeatedly multiplying the matrix with itself.
How would you scale the matrix down to a, say monthly, transition matrix?
I found a result on reversible Markov Chains and thought the following:
Let $M$ be the transition matrix of a reversible markov chain, then one can factor it into $$ M = S D$$
with a symmetric matrix $S$ and a diagonal matrix $D$. It appears that also the converse holds true.
Now, if we had a reversible markov chain, we could raise the matrix to a fraction by taking the eigenvalue decomposition of $S = E^{T}\Lambda E$ and raising $S$ as well as $D$ to the appropriate power.
So we define
$$ \bar{M}:= E^T \Lambda^{\frac{1}{n}}E D^{\frac{1}{n}}$$ and say that $\bar{M}$ is now $M^{\frac{1}{n}}$.
The question is: Is this way appropriate? Does it make any sense (from a mathematical viewpoint - I know that downscaling a transition matrix is debatable)? Are ratings transition matrices from a reversible markov chain?
The main question is: What are the commonly used method to raise ratings transition matrices to a fraction?