Bohemian Matrices

UH_1_Subdiag_0_1

Upper Hessenberg matrices with subdiagonal entries fixed at 1 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 (72B) CharPolys_2x2.zip (226B) CharPolys_2x2.tar.gz (174B)
3x3 CharPolys_3x3.csv (348B) CharPolys_3x3.zip (303B) CharPolys_3x3.tar.gz (259B)
4x4 CharPolys_4x4.csv (3KB) CharPolys_4x4.zip (831B) CharPolys_4x4.tar.gz (806B)
5x5 CharPolys_5x5.csv (37KB) CharPolys_5x5.zip (8KB) CharPolys_5x5.tar.gz (8KB)
6x6 CharPolys_6x6.csv (769KB) CharPolys_6x6.zip (155KB) CharPolys_6x6.tar.gz (155KB)
7x7 CharPolys_7x7.csv (25MB) CharPolys_7x7.zip (5MB) CharPolys_7x7.tar.gz (5MB)
8x8 CharPolys_8x8.csv (1GB) CharPolys_8x8.zip (237MB) CharPolys_8x8.tar.gz (237MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (72B) MinPolys_2x2.zip (224B) MinPolys_2x2.tar.gz (172B)
3x3 MinPolys_3x3.csv (348B) MinPolys_3x3.zip (301B) MinPolys_3x3.tar.gz (257B)
4x4 MinPolys_4x4.csv (3KB) MinPolys_4x4.zip (829B) MinPolys_4x4.tar.gz (802B)
5x5 MinPolys_5x5.csv (37KB) MinPolys_5x5.zip (8KB) MinPolys_5x5.tar.gz (8KB)
6x6 MinPolys_6x6.csv (769KB) MinPolys_6x6.zip (155KB) MinPolys_6x6.tar.gz (155KB)
7x7 MinPolys_7x7.csv (25MB) MinPolys_7x7.zip (5MB) MinPolys_7x7.tar.gz (5MB)
8x8 MinPolys_8x8.csv (1GB) MinPolys_8x8.zip (237MB) MinPolys_8x8.tar.gz (237MB)

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 8 6 6 8 2
3x3 64 28 28 64 3
4x4 1,024 197 197 1,024 6
5x5 32,768 2,235 2,235 32,768 13
6x6 2,097,152 39,768 39,768 2,097,152 27
7x7 268,435,456 1,140,848     55
8x8 68,719,476,736 51,925,106     120

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 6 6 0 6
3x3 43 25 18 28
4x4 413 219 194 197
5x5 6,920 3,264 3,656 2,235
6x6 166,005 75,045 90,960 39,768
7x7 6,176,985 2,694,199 3,482,786  
8x8        

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
OEIS                    
2x2 4 4 4 4            
3x3 30 34 0 30 34          
4x4 440 584 0 0 440 584        
5x5 12,792 19,976 0 0 0 12,792 19,976      
6x6 740,732 1,356,420 0 0 0 0 740,732 1,356,420    
7x7 85,591,032 182,844,424                
8x8 19,751,265,584 48,968,211,152                

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS   A000045  
2x2 2 1 4
3x3 5 2 32
4x4 7 3 508
5x5 11 5 15,944
6x6 17 8 988,836
7x7 27 13 121,294,672
8x8 43 21 29,467,080,660

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