Bohemian Matrices

UH_1_Subdiag_0_Diag_0_1

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

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (35B) CharPolys_2x2.zip (212B) CharPolys_2x2.tar.gz (161B)
3x3 CharPolys_3x3.csv (89B) CharPolys_3x3.zip (235B) CharPolys_3x3.tar.gz (180B)
4x4 CharPolys_4x4.csv (398B) CharPolys_4x4.zip (300B) CharPolys_4x4.tar.gz (253B)
5x5 CharPolys_5x5.csv (3KB) CharPolys_5x5.zip (784B) CharPolys_5x5.tar.gz (757B)
6x6 CharPolys_6x6.csv (35KB) CharPolys_6x6.zip (6KB) CharPolys_6x6.tar.gz (6KB)
7x7 CharPolys_7x7.csv (626KB) CharPolys_7x7.zip (109KB) CharPolys_7x7.tar.gz (109KB)
8x8 CharPolys_8x8.csv (17MB) CharPolys_8x8.zip (3MB) CharPolys_8x8.tar.gz (3MB)
9x9 CharPolys_9x9.csv (717MB) CharPolys_9x9.zip (114MB) CharPolys_9x9.tar.gz (114MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (35B) MinPolys_2x2.zip (210B) MinPolys_2x2.tar.gz (159B)
3x3 MinPolys_3x3.csv (89B) MinPolys_3x3.zip (233B) MinPolys_3x3.tar.gz (179B)
4x4 MinPolys_4x4.csv (398B) MinPolys_4x4.zip (298B) MinPolys_4x4.tar.gz (251B)
5x5 MinPolys_5x5.csv (3KB) MinPolys_5x5.zip (782B) MinPolys_5x5.tar.gz (755B)
6x6 MinPolys_6x6.csv (35KB) MinPolys_6x6.zip (6KB) MinPolys_6x6.tar.gz (6KB)
7x7 MinPolys_7x7.csv (626KB) MinPolys_7x7.zip (109KB) MinPolys_7x7.tar.gz (109KB)
8x8 MinPolys_8x8.csv (17MB) MinPolys_8x8.zip (3MB) MinPolys_8x8.tar.gz (3MB)
9x9 MinPolys_9x9.csv (717MB) MinPolys_9x9.zip (114MB) MinPolys_9x9.tar.gz (114MB)

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 A006125        
2x2 2 2 2 2 1
3x3 8 6 6 8 2
4x4 64 27 27 64 3
5x5 1,024 180 180 1,024 4
6x6 32,768 1,863 1,863 32,768 6
7x7 2,097,152 29,193 29,193 2,097,152 12
8x8 268,435,456 711,662     20
9x9 68,719,476,736 26,708,805     33

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 3 3 0 2
3x3 12 8 4 6
4x4 72 40 32 27
5x5 661 311 350 180
6x6 8,759 3,865 4,894 1,863
7x7 173,506 73,994 99,512 29,193
8x8 5,115,981 2,126,127 2,989,854  
9x9        

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 Number of Rank 8 Matrices Number of Rank 9 Matrices
OEIS                      
2x2 1 1 1 1              
3x3 4 4 0 4 4            
4x4 32 32 0 0 32 32          
5x5 480 544 0 0 0 480 544        
6x6 14,080 18,688 0 0 0 0 14,080 18,688      
7x7 822,272 1,274,880 0 0 0 0 0 822,272 1,274,880    
8x8 95,969,280 172,466,176                  
9x9 22,383,869,952 46,335,606,784                  

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 2 1 1
3x3 2 1 4
4x4 3 1 32
5x5 4 2 512
6x6 6 3 16,256
7x7 9 4 1,024,000
8x8 12 6 127,795,200
9x9 18 9 31,583,748,096

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     1      
4x4     1      
5x5     1      
6x6     1      
7x7     1      
8x8     1      
9x9     1