Bohemian Matrices

UH_1_Subdiag_n1_1

Upper Hessenberg matrices with subdiagonal entries fixed at 1 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 (69B) CharPolys_2x2.zip (230B) CharPolys_2x2.tar.gz (176B)
3x3 CharPolys_3x3.csv (383B) CharPolys_3x3.zip (312B) CharPolys_3x3.tar.gz (266B)
4x4 CharPolys_4x4.csv (4KB) CharPolys_4x4.zip (1KB) CharPolys_4x4.tar.gz (1KB)
5x5 CharPolys_5x5.csv (82KB) CharPolys_5x5.zip (16KB) CharPolys_5x5.tar.gz (16KB)
6x6 CharPolys_6x6.csv (3MB) CharPolys_6x6.zip (543KB) CharPolys_6x6.tar.gz (543KB)
7x7 CharPolys_7x7.csv (237MB) CharPolys_7x7.zip (37MB) CharPolys_7x7.tar.gz (37MB)
8x8 CharPolys_8x8.csv (33GB) CharPolys_8x8.zip (5GB) CharPolys_8x8.tar.gz (5GB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (69B) MinPolys_2x2.zip (228B) MinPolys_2x2.tar.gz (173B)
3x3 MinPolys_3x3.csv (383B) MinPolys_3x3.zip (310B) MinPolys_3x3.tar.gz (263B)
4x4 MinPolys_4x4.csv (4KB) MinPolys_4x4.zip (1KB) MinPolys_4x4.tar.gz (1KB)
5x5 MinPolys_5x5.csv (82KB) MinPolys_5x5.zip (16KB) MinPolys_5x5.tar.gz (16KB)
6x6 MinPolys_6x6.csv (3MB) MinPolys_6x6.zip (543KB) MinPolys_6x6.tar.gz (543KB)
7x7 MinPolys_7x7.csv (237MB) MinPolys_7x7.zip (37MB) MinPolys_7x7.tar.gz (37MB)
8x8 MinPolys_8x8.csv (33GB) MinPolys_8x8.zip (5GB) MinPolys_8x8.tar.gz (5GB)

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 32 32 64 5
4x4 1,024 289 289 1,024 12
5x5 32,768 4,958 4,958 32,768 28
6x6 2,097,152 162,059 162,059 2,097,152 66
7x7 268,435,456 10,318,948     168
8x8 68,719,476,736 1,279,227,671     416

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 9 5 4 6
3x3 65 29 36 32
4x4 689 233 456 289
5x5 20,565 7,363 13,202 4,958
6x6 887,539 299,477 588,062 162,059
7x7        
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 24 40 0 24 40          
4x4 304 720 0 0 304 720        
5x5 7,648 25,120 0 0 0 7,648 25,120      
6x6 383,680 1,713,472 0 0 0 0 383,680 1,713,472    
7x7 38,422,400 230,013,056                
8x8 7,683,485,440 61,035,991,296                

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS   A000079  
2x2 3 2 0
3x3 5 4 0
4x4 9 8 0
5x5 17 16 0
6x6 33 32 0
7x7 65 64 0
8x8 129 128 0

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     2      
3x3     0      
4x4     0      
5x5     0      
6x6     324      
7x7     0      
8x8     0