Bohemian Matrices

UH_1_Subdiag_0_1_2

Upper Hessenberg matrices with subdiagonal entries fixed at 1 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 (165B) CharPolys_2x2.zip (256B) CharPolys_2x2.tar.gz (203B)
3x3 CharPolys_3x3.csv (2KB) CharPolys_3x3.zip (725B) CharPolys_3x3.tar.gz (699B)
4x4 CharPolys_4x4.csv (53KB) CharPolys_4x4.zip (12KB) CharPolys_4x4.tar.gz (12KB)
5x5 CharPolys_5x5.csv (3MB) CharPolys_5x5.zip (625KB) CharPolys_5x5.tar.gz (625KB)
6x6 CharPolys_6x6.csv (309MB) CharPolys_6x6.zip (61MB) CharPolys_6x6.tar.gz (61MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (165B) MinPolys_2x2.zip (254B) MinPolys_2x2.tar.gz (203B)
3x3 MinPolys_3x3.csv (2KB) MinPolys_3x3.zip (723B) MinPolys_3x3.tar.gz (698B)
4x4 MinPolys_4x4.csv (53KB) MinPolys_4x4.zip (12KB) MinPolys_4x4.tar.gz (12KB)
5x5 MinPolys_5x5.csv (3MB) MinPolys_5x5.zip (625KB) MinPolys_5x5.tar.gz (625KB)
6x6 MinPolys_6x6.csv (309MB) MinPolys_6x6.zip (61MB) MinPolys_6x6.tar.gz (61MB)

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 27 16 16 27 4
3x3 729 170 170 729 12
4x4 59,049 3,584 3,584 59,049 36
5x5 14,348,907 157,484 157,484 14,348,907 104
6x6 10,460,353,203 14,072,442     292

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 17 17 0 16
3x3 310 210 100 170
4x4 10,157 6,331 3,826 3,584
5x5 639,021 373,367 265,654 157,484
6x6        

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
OEIS                
2x2 8 19            
3x3 171 558            
4x4 10,772 48,277            
5x5 2,030,404 12,318,503            
6x6 1,149,241,743 9,311,111,460            

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS      
2x2 7 4 9
3x3 19 10 216
4x4 43 24 15,107
5x5 103 58 3,102,718
6x6 247 140 1,878,883,473

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