Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic).

We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we have:

$\| APA^{-1} \|\_2 \leq 1$

Here $ \| \cdot \|\_2$ means a matrix norm induced by the vector norm $L\_2$.

One example is $\operatorname{diag}(\xi)^{1/2}$, where $\xi$ is the vector representing the stationary distribution and $\operatorname{diag}$ is an operator constructing a diagonal matrix with zeros off the diagonal.

I am interested in a rule defining all (or at least a large class) of valid choices for $A$.