Projects › Bohemian Matrices
Visualizing Eigenvalues of Random Integer Matrices
Bohemian matrices are families of random matrices whose entries are drawn from a fixed, finite set of integers. Plotting all eigenvalues of an entire matrix family on the complex plane reveals intricate, structured, and often beautiful geometric patterns.
What are Bohemian Matrices?
The name is derived from BOunded HEight Matrix of Integers (BOHEMI). A Bohemian family consists of all matrices of a given structure — such as upper Hessenberg or Toeplitz — whose entries are drawn from a predetermined finite discrete set, called the population. Typical populations are small sets like {−1, 0, 1} or {0, 1}.
Even for modest dimensions, a family can contain billions of matrices. Their collective eigenvalue spectra, plotted in the complex plane and colored by density, form striking and geometrically rich images. Characteristic features include sharp exclusion zones — regions where no eigenvalues can ever fall — and dense fractal-like structures.
The BHIME Project
The BHIME (Bounded Height Integer Matrix Eigenvalues) project is a MATLAB toolkit developed by Steven Thornton and Robert Corless at the University of Western Ontario for generating high-resolution eigenvalue density images. It uses a three-stage pipeline: random sampling of matrices from the family, accumulation of eigenvalues onto a fine complex-plane grid, and rendering with customizable colormaps.
The images here were produced using this toolkit, exploring different matrix structures and populations.
Eigenvalue Visualizations
Resources
Bohemian Matrices Website
The main project hub, hosting an image gallery, database of computed families, and research background.
BHIME MATLAB Toolkit
Open-source MATLAB code for sampling Bohemian families, processing eigenvalue data, and rendering density images.
