{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-fractal-eigenvector","title":"A Fractal Eigenvector","arxiv_id":"2104.01116","date":"2021-03-29","proceeding":null,"authors":["Neil J. Calkin","Eunice Y. S. Chan","Robert M. Corless","David J. Jeffrey","Piers W. Lawrence"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2104.01116v1","url_pdf":"https://arxiv.org/pdf/2104.01116v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-fractal-eigenvector","repo_url":"https://github.com/rcorless/FractalEigenvector","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}