Bohemian Matrices

UH_1_Subdiag_0_Diag_n1_0

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

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (34B) CharPolys_2x2.zip (210B) CharPolys_2x2.tar.gz (159B)
3x3 CharPolys_3x3.csv (82B) CharPolys_3x3.zip (228B) CharPolys_3x3.tar.gz (176B)
4x4 CharPolys_4x4.csv (350B) CharPolys_4x4.zip (293B) CharPolys_4x4.tar.gz (246B)
5x5 CharPolys_5x5.csv (3KB) CharPolys_5x5.zip (724B) CharPolys_5x5.tar.gz (699B)
6x6 CharPolys_6x6.csv (30KB) CharPolys_6x6.zip (6KB) CharPolys_6x6.tar.gz (6KB)
7x7 CharPolys_7x7.csv (545KB) CharPolys_7x7.zip (103KB) CharPolys_7x7.tar.gz (103KB)
8x8 CharPolys_8x8.csv (16MB) CharPolys_8x8.zip (3MB) CharPolys_8x8.tar.gz (3MB)
9x9 CharPolys_9x9.csv (732MB) CharPolys_9x9.zip (120MB) CharPolys_9x9.tar.gz (120MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (34B) MinPolys_2x2.zip (208B) MinPolys_2x2.tar.gz (157B)
3x3 MinPolys_3x3.csv (82B) MinPolys_3x3.zip (226B) MinPolys_3x3.tar.gz (173B)
4x4 MinPolys_4x4.csv (350B) MinPolys_4x4.zip (291B) MinPolys_4x4.tar.gz (242B)
5x5 MinPolys_5x5.csv (3KB) MinPolys_5x5.zip (722B) MinPolys_5x5.tar.gz (696B)
6x6 MinPolys_6x6.csv (30KB) MinPolys_6x6.zip (6KB) MinPolys_6x6.tar.gz (6KB)
7x7 MinPolys_7x7.csv (545KB) MinPolys_7x7.zip (103KB) MinPolys_7x7.tar.gz (103KB)
8x8 MinPolys_8x8.csv (16MB) MinPolys_8x8.zip (3MB) MinPolys_8x8.tar.gz (3MB)
9x9 MinPolys_9x9.csv (732MB) MinPolys_9x9.zip (120MB) MinPolys_9x9.tar.gz (120MB)

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 24 5
6x6 32,768 1,849 1,849 32,768 9
7x7 2,097,152 29,223 29,223 2,097,152 15
8x8 268,435,456 740,780     31
9x9 68,719,476,736 28,870,492     57

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 1 2 2
3x3 14 4 10 6
4x4 79 7 72 27
5x5 699 109 590 180
6x6 9,389 789 8,600 1,849
7x7 184,137 24,061 160,076 29,223
8x8 5,599,761 501,149 5,098,612  
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 24 40 0 0 24 40          
5x5 288 736 0 0 0 288 736        
6x6 6,528 26,240 0 0 0 0 6,528 26,240      
7x7 283,136 1,814,016 0 0 0 0 0 283,136 1,814,016    
8x8 23,574,528 244,860,928                  
9x9 3,788,185,600 64,931,291,136                  

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 2 1 1
3x3 2 1 4
4x4 3 2 32
5x5 4 3 480
6x6 6 5 12,928
7x7 9 8 643,072
8x8 14 13 59,578,368
9x9 22 21 10,401,914,880

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