A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each can be called a binary matrix) eigenvalues whose negatives $-\lambda = c$ give periodic orbits under the iteration, namely $z_k z_{k-1}^2+c$ with $z_0=0$, are thus contained in set. By Perron--Frobenius theorem, dominant real positive eigenvalue, which we call $\rho_n$. This article examines eigenvector belonging to that eigenvalue its fractal-like structure, similarly (with less success) singular vectors from value decomposition.