Let $M \in \mathbb R_+^{N,N}$ be a nonnegative stochastic matrix and symmetric w.r.t some scalar product $\langle \cdot \, \vert \, \cdot \rangle_s$ given by some positive vector $s \in \mathbb R_*^{N}$.
On page 36 in https://tel.archives-ouvertes.fr/tel-02926037/document I then read that we have
$$M^Ts=s$$ which according to the paper comes from $M \textbf{1}=\textbf{1}$ and the symmetry of $d(s)M$, where $d(s)$ is the diagonal matrix with components of $s$ on the diagonal. I have a too foggy view to actually see how we from that get $M^Ts=s.$ Can anyone help me there? And what does it exactly mean, that the matrix is symmetric w.r.t the scalar product? Havent dealt alot with matrix scalar products so far.
That "$M$ is symmetric with respect to the scalar product $\langle \cdot,\cdot\rangle_s$ given by a positive vector $s$" means that $M^T\operatorname{diag}(s)=\operatorname{diag}(s)M$. So, when $M$ is row-stochastic, $$ M^Ts =M^T\operatorname{diag}(s)\mathbf1 =\operatorname{diag}(s)M\mathbf1 =\operatorname{diag}(s)\mathbf1 =s. $$