A Fractal Eigenvector

Abstract

The recursively-constructed family of Mandelbrot matrices for n = 1, 2, have nonnegative entries (indeed just 0 and 1, so each can be called a binary matrix) and have eigenvalues whose negatives give periodic orbits under the Mandelbrot iteration, namely with , and are thus contained in the Mandelbrot set. By the Perron–Frobenius theorem, the matrices have a dominant real positive eigenvalue, which we call ρn . 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 from the singular value decomposition.