Bohemian Matrices

UH_1_Subdiag_0_Diag_0_1_2

Upper Hessenberg matrices with subdiagonal entries fixed at 1, diagonal entries fixed at 0, and all other entries from the set {0, +1, +2}.

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (44B) CharPolys_2x2.zip (215B) CharPolys_2x2.tar.gz (163B)
3x3 CharPolys_3x3.csv (194B) CharPolys_3x3.zip (257B) CharPolys_3x3.tar.gz (205B)
4x4 CharPolys_4x4.csv (2KB) CharPolys_4x4.zip (626B) CharPolys_4x4.tar.gz (597B)
5x5 CharPolys_5x5.csv (43KB) CharPolys_5x5.zip (8KB) CharPolys_5x5.tar.gz (8KB)
6x6 CharPolys_6x6.csv (2MB) CharPolys_6x6.zip (338KB) CharPolys_6x6.tar.gz (338KB)
7x7 CharPolys_7x7.csv (135MB) CharPolys_7x7.zip (24MB) CharPolys_7x7.tar.gz (24MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (44B) MinPolys_2x2.zip (213B) MinPolys_2x2.tar.gz (162B)
3x3 MinPolys_3x3.csv (194B) MinPolys_3x3.zip (255B) MinPolys_3x3.tar.gz (200B)
4x4 MinPolys_4x4.csv (2KB) MinPolys_4x4.zip (624B) MinPolys_4x4.tar.gz (594B)
5x5 MinPolys_5x5.csv (43KB) MinPolys_5x5.zip (8KB) MinPolys_5x5.tar.gz (8KB)
6x6 MinPolys_6x6.csv (2MB) MinPolys_6x6.zip (338KB) MinPolys_6x6.tar.gz (338KB)
7x7 MinPolys_7x7.csv (135MB) MinPolys_7x7.zip (24MB) MinPolys_7x7.tar.gz (24MB)

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 2
3x3 27 15 15 27 4
4x4 729 150 150 729 6
5x5 59,049 2,559 2,559 59,049 12
6x6 14,348,907 90,087 90,087 14,348,907 24
7x7 10,460,353,203 5,766,951     48

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 5 0 3
3x3 34 24 10 15
4x4 478 290 188 150
5x5 10,988 6,266 4,722 2,559
6x6 496,612 271,770 224,842 90,087
7x7        

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 216 513 0 0 216 513      
5x5 13,608 45,441 0 0 0 13,608 45,441    
6x6 2,624,400 11,724,507 0 0 0 0 2,624,400 11,724,507  
7x7 1,535,116,536 8,925,236,667              

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 3 2 1
3x3 3 2 9
4x4 7 4 243
5x5 11 8 17,010
6x6 23 12 3,641,355
7x7 43 26 2,327,101,407

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