Bohemian Matrices

UH_1_Subdiag_0_Diag_n1_0_1

Upper Hessenberg matrices with subdiagonal entries fixed at 1, diagonal entries fixed at 0, and all other entries from the set {-1, 0, +1}.

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (43B) CharPolys_2x2.zip (214B) CharPolys_2x2.tar.gz (163B)
3x3 CharPolys_3x3.csv (183B) CharPolys_3x3.zip (257B) CharPolys_3x3.tar.gz (201B)
4x4 CharPolys_4x4.csv (2KB) CharPolys_4x4.zip (572B) CharPolys_4x4.tar.gz (540B)
5x5 CharPolys_5x5.csv (36KB) CharPolys_5x5.zip (7KB) CharPolys_5x5.tar.gz (7KB)
6x6 CharPolys_6x6.csv (1MB) CharPolys_6x6.zip (239KB) CharPolys_6x6.tar.gz (239KB)
7x7 CharPolys_7x7.csv (77MB) CharPolys_7x7.zip (14MB) CharPolys_7x7.tar.gz (14MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (43B) MinPolys_2x2.zip (212B) MinPolys_2x2.tar.gz (161B)
3x3 MinPolys_3x3.csv (183B) MinPolys_3x3.zip (255B) MinPolys_3x3.tar.gz (200B)
4x4 MinPolys_4x4.csv (2KB) MinPolys_4x4.zip (570B) MinPolys_4x4.tar.gz (536B)
5x5 MinPolys_5x5.csv (36KB) MinPolys_5x5.zip (7KB) MinPolys_5x5.tar.gz (7KB)
6x6 MinPolys_6x6.csv (1MB) MinPolys_6x6.zip (239KB) MinPolys_6x6.tar.gz (239KB)
7x7 MinPolys_7x7.csv (77MB) MinPolys_7x7.zip (14MB) MinPolys_7x7.tar.gz (14MB)

Properties

Download all properties: Properties.csv

Characteristic Polynomials

Matrix Size Number of Matrices Number of Characteristic Polynomials Number of Minimal Polynomials Number of Non-Derogatory Matrices Maximum Characteristic Height
OEIS A047656        
2x2 3 3 3 3 1
3x3 27 15 15 27 2
4x4 729 140 140 729 4
5x5 59,049 2,297 2,297 59,049 5
6x6 14,348,907 67,628 67,628 14,348,907 9
7x7 10,460,353,203 3,606,225     15

Eigenvalues

Matrix Size Number of Distinct Eigenvalues Number of Distinct Real Eigenvalues Number of Distinct Purely Complex Eigenvalues Number of Distinct Jordan Canonical Forms
OEIS        
2x2 5 3 2 3
3x3 35 15 20 15
4x4 431 135 296 140
5x5 9,497 2,877 6,620 2,297
6x6 363,143 103,987 259,156 67,628
7x7 23,795,195 6,648,959 17,146,236  

Rank

Matrix Size Number of Singular Matrices Number of Non-Singular Matrices Number of Rank 1 Matrices Number of Rank 2 Matrices Number of Rank 3 Matrices Number of Rank 4 Matrices Number of Rank 5 Matrices Number of Rank 6 Matrices Number of Rank 7 Matrices
OEIS                  
2x2 1 2 1 2          
3x3 9 18 0 9 18        
4x4 243 486 0 0 243 486      
5x5 17,739 41,310 0 0 0 17,739 41,310    
6x6 3,733,209 10,615,698 0 0 0 0 3,733,209 10,615,698  
7x7 2,334,620,313 8,125,732,890              

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 3 1 2
3x3 3 1 18
4x4 5 2 378
5x5 7 3 27,702
6x6 11 5 6,075,486
7x7 17 8 3,965,691,474

Miscellaneous

Matrix Size Number of Normal Matrices Number of Rhapsodic Matrices Number of Nilpotent Matrices Number of Totally Unimodular Matrices Number of Type I Stable Matrices Number of Type II Stable Matrices
OEIS            
2x2     1      
3x3     3      
4x4     21      
5x5     271      
6x6     9,075      
7x7     685,033