A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices M-n for n = 1, 2,... 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.