A Fractal Eigenvector
Neil J. Calkin, Eunice Y. S. Chan, Robert M. Corless, David J. Jeffrey, Piers W. Lawrence · American Mathematical Monthly · 2022
The recursively-constructed family of Mandelbrot matrices Mn for n = 1, 2, … have nonnegative entries (indeed just 0 and 1, so each Mn can be called a binary matrix) and have eigenvalues whose negatives −λ=c give periodic orbits under the Mandelbrot iteration, namely zk=zk−12+c with z0=0, and are thus contained in the Mandelbrot set. By the Perron–Frobenius theorem, the matrices Mn 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 Mn from the singular value decomposition.