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