In the similarity transformation $$\cases{{\bf A = T}^{-1}{\bf DT}\\ {\bf D}_{ij} = \lambda_j\delta_{i-j}}$$
The eigenvalues $\lambda_j$ of $\bf A$ are on the diagonal in $\bf D$, and corresponding eigenvectors are the columns in $\bf T$. For some $\bf A$ matrices we can't find such a pair of $\bf T,D$ but instead we can get a block diagonal decomposition. Typical things that can affect if we can find such a similarity is what kinds of numbers we allow the matrix elements to be. If we allow complex numbers, which are in some sense more complicated than real numbers, then we can diagonalize more matrices than if we only allow real numbers. But if we allow blocks of real numbers on the diagonal we can once again find similarity.
Now to the question: In the spirit of this question, can we go in the opposite direction, disallowing negative reals (or replace them with 2x2 blocks of non-negatives like in the other questions) and be able to create a block-diagonal "eigenvalue" decomposition? Or rather, under which circumstances (properties of $\bf A$ et.c.) would we be able to do it?
If we could characterize it. Maybe we would be able to define some kind of canonical basis for diffusion or random walks.