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?



Similarity transformation of transition matrix of reversible Markov chain (reference request)
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.](http://papers.nips.cc/paper/2665-hierarchical-eigensolver-for-transition-matrices-in-spectral-methods.pdf) Other papers simply call $R$ the symmetrized transition matrix, see for example [here.](http://arxiv.org/abs/1511.08100)


