Bohemian Matrices

UH_1_Subdiag_0_Diag_n1_1

Upper Hessenberg matrices with subdiagonal entries fixed at 1, diagonal entries fixed at 0, and all other entries from the set {-1, +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 (87B) CharPolys_3x3.zip (233B) CharPolys_3x3.tar.gz (179B)
4x4 CharPolys_4x4.csv (417B) CharPolys_4x4.zip (301B) CharPolys_4x4.tar.gz (253B)
5x5 CharPolys_5x5.csv (4KB) CharPolys_5x5.zip (862B) CharPolys_5x5.tar.gz (832B)
6x6 CharPolys_6x6.csv (56KB) CharPolys_6x6.zip (9KB) CharPolys_6x6.tar.gz (9KB)
7x7 CharPolys_7x7.csv (1MB) CharPolys_7x7.zip (247KB) CharPolys_7x7.tar.gz (247KB)
8x8 CharPolys_8x8.csv (70MB) CharPolys_8x8.zip (11MB) CharPolys_8x8.tar.gz (11MB)
9x9 CharPolys_9x9.csv (6GB) CharPolys_9x9.zip (823MB) CharPolys_9x9.tar.gz (823MB)

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 (87B) MinPolys_3x3.zip (231B) MinPolys_3x3.tar.gz (177B)
4x4 MinPolys_4x4.csv (417B) MinPolys_4x4.zip (299B) MinPolys_4x4.tar.gz (253B)
5x5 MinPolys_5x5.csv (4KB) MinPolys_5x5.zip (860B) MinPolys_5x5.tar.gz (831B)
6x6 MinPolys_6x6.csv (56KB) MinPolys_6x6.zip (9KB) MinPolys_6x6.tar.gz (9KB)
7x7 MinPolys_7x7.csv (1MB) MinPolys_7x7.zip (247KB) MinPolys_7x7.tar.gz (247KB)
8x8 MinPolys_8x8.csv (70MB) MinPolys_8x8.zip (11MB) MinPolys_8x8.tar.gz (11MB)
9x9 MinPolys_9x9.csv (6GB) MinPolys_9x9.zip (823MB) MinPolys_9x9.tar.gz (823MB)

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 30 30 64 3
5x5 1,024 232 232 1,024 5
6x6 32,768 3,112 3,112 32,768 9
7x7 2,097,152 72,574 72,574 2,097,152 15
8x8 268,435,456 2,946,276     31
9x9 68,719,476,736 209,900,412     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 4 2 2 2
3x3 16 8 8 6
4x4 81 23 58 30
5x5 962 298 664 232
6x6 16,982 4,948 12,034 3,112
7x7 460,307 125,403 334,904 72,574
8x8        
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 0 2 0 2              
3x3 0 8 0 0 8            
4x4 32 32 0 0 32 32          
5x5 0 1,024 0 0 0 0 1,024        
6x6 0 32,768 0 0 0 0 0 32,768      
7x7 573,440 1,523,712 0 0 0 0 0 573,440 1,523,712    
8x8 0 268,435,456                  
9x9 0 68,719,476,736                  

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 2 1 2
3x3 2 1 8
4x4 3 2 0
5x5 4 3 768
6x6 6 5 20,480
7x7 9 8 0
8x8 14 13 114,819,072
9x9 22 21 24,373,100,544

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     0      
3x3     0      
4x4     0      
5x5     0      
6x6     0      
7x7     0      
8x8     0      
9x9     0