Characteristic Polynomial Database

The Characteristic Polynomial Database should be accessed through the Bohemian Matrices website at BohemianMatrices.com/cpdb. Data has been removed from the repository.

This repository contains characteristic polynomials and properties of a variety of families of Bohemian matrices. All source files are available within the repository for reproducing the results. Currently the database contains 1,762,728,065 characteristic polynomials from 2,366,960,967,336 matrices.

Bohemian Families

Below is the summary of the families of Bohemian matrices currently available in the repository. We are actively adding more families and more properties.

Unstructured

Matrices with no structure. For example, 5x5 matrices with entries from the set {-1, 0, +1}.

Upper Hessenberg

Upper Hessenberg matrices with subdiagonal entries fixed at 1.

Upper Hessenberg Toeplitz

Coming Soon! Upper Hessenberg matrices with a Toeplitz structure with subdiagonal entries fixed at either -1 or 1.

Data Files

The files containing characteristic polynomials are CSV files with no header. They all follow the naming convention CharPolys_nxn.csv where n is the dimension of the matrix they come from. Each row within the CSV files is of the form

count, c_{n-1}, c_{n-2}, ..., c_0

where count is the number of times the characteristic polynomial appears in the family. c_{n-1} to c_0 are the coefficients of the characteristic polynomial beginning with the degree n-1 coefficient down to the constant coefficient. The leading coefficient (c_n) is omitted from the files as this will always be equal to 1.

A Maple function read_poly_data has been provided in the src/Maple directory for loading the polynomials (and optionally their counts) into Maple. See the header in the file for details on its usage.

To Do

• Add families
  • Upper Hessenberg Toeplitz
  • Symmetric
  • Upper Triangular
  • Circulant
  • Tridiagonal
  • Toeplitz
  • Hankel
• Convert Maple scripts to SymPy
• Add plots of eigenvalues
• Add random samples of characteristic polynomials for large classes
• Add sequences to OEIS
• Add leading coefficient to characteristic polynomials
• Add gcd-free bases sets
• Add sets of invariant factors
• Add properties:
  • Smallest positive integer that is not a determinant (see A013588)
  • Number of different values taken by permanent of matrix (see A087983)
  • Number of matrices such that all eigenvalues are real
  • Number of matrices such that some eigenvalues are strictly complex (see A098148)
  • Number of matrices with det(A) = 1 (see A086264)
  • Number of matrices that are diagonalizable over the complex numbers
  • Number of positive definite matrices
  • Number of matrices with real and positive eigenvalues
  • Number of primitive matrices
  • Number of matrices with 0 permanent
  • Number of idenpotent matrices (eigenvalues are all roots of unity)
  • Number of neutral matrices (matrices whose eigenvalues all have 0 real part)
  • Number of neutral characteristic polynomials (polynomials whose roots all have 0 real part)

Unstructured

• Compute properties:
  • Number of Distinct Jordan Canonical Forms
  • Number of Normal Matrices
  • Number of Rhapsodic Matrices
  • Number of Totally Unimodular Matrices
  • Number of Type I Stable Matrices
  • Number of Type II Stable Matrices

Upper Hessenberg

• Compute properties:
  • Number of Distinct Jordan Canonical Forms
  • Number of Normal Matrices
  • Number of Rhapsodic Matrices
  • Number of Totally Unimodular Matrices
  • Number of Type I Stable Matrices
  • Number of Type II Stable Matrices