$\rho(A)^{-1}D^{-1}AD$ a stochastic matrix with $A > 0 \in M_n$?

Question

Nonnegative Matrix Theory Experts,

Can we prove that $\rho(A)^{-1}D^{-1}AD$ is a stochastic matrix?

• $A > 0 \in M_n$ is a non-negative non-zero matrix that has a positive eigenvector say $x = [x_1,\cdots,x_n]$,
• $D = \textrm{diag}\left(x_1,\cdots,x_n\right)$ is a diagonal matrix comprising the elements of the eigenvector $x$ of $A$,
• and $\rho(A)$ is a spectral radius of $A$.

Thank you so much in advance,

Answer 1

It's a duplicate: there is a same question with some answers herein If $A\ge0$, $A$ has a positive eigenvector, $D = diag(x_1, . . . , x_n).$ $ \Rightarrow $ $\rho {(A)^{ - 1}}({D^{ - 1}}AD)$ is stochastic