Similarity transformation of transition matrix of reversible Markov chain (reference request)

Question

If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by:

$$R_{ij} = \sqrt{\frac{\pi_i}{\pi_j}}P_{ij}~.$$

By reversibility, it follows that $R$ is a symmetric matrix arising from a similarity transformation of $P$ and hence, both $P$ and $R$ have real eigenvalues.

Has this important matrix $R$ been given a name in the literature?

Answer 1

If $P$ is the Markov transition matrix of a graph, the matrix $A=P\pi^{-1}$ is called the affinity matrix and $R=\pi^{1/2}A\pi^{1/2}=\pi^{1/2}P\pi^{-1/2}$ is called the normalized affinity matrix, see for example section 3 of this paper. Other papers simply call $R$ the symmetrized transition matrix, see for example here.