is there any relation between the right eigenvectors of $P^\top$ and $\begin{bmatrix}0 &P^\top \\P &0\end{bmatrix}$?

Question

I'm trying to analyze the stationary distribution of a Markov Chain parameterized by some n by n matrix $P$ (top right eigenvector of $P^\top$). While there is a relation between the singular vectors of $P^\top$ and those of $\begin{bmatrix}0 &P^\top\\P &0\end{bmatrix}$. However, I couldn't find anything on eigenvectors.

Also, $P\mathbf{1} = 1$ (row sum is 1) and that $P$ is not symmetric.

Any potential leads will be much appreciated. Thanks in advance.