Bohemian Matrices

UH_1_Subdiag_n1_0_1

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

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (156B) CharPolys_2x2.zip (256B) CharPolys_2x2.tar.gz (205B)
3x3 CharPolys_3x3.csv (2KB) CharPolys_3x3.zip (679B) CharPolys_3x3.tar.gz (652B)
4x4 CharPolys_4x4.csv (46KB) CharPolys_4x4.zip (10KB) CharPolys_4x4.tar.gz (10KB)
5x5 CharPolys_5x5.csv (2MB) CharPolys_5x5.zip (473KB) CharPolys_5x5.tar.gz (473KB)
6x6 CharPolys_6x6.csv (215MB) CharPolys_6x6.zip (41MB) CharPolys_6x6.tar.gz (41MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (156B) MinPolys_2x2.zip (254B) MinPolys_2x2.tar.gz (203B)
3x3 MinPolys_3x3.csv (2KB) MinPolys_3x3.zip (677B) MinPolys_3x3.tar.gz (650B)
4x4 MinPolys_4x4.csv (46KB) MinPolys_4x4.zip (10KB) MinPolys_4x4.tar.gz (10KB)
5x5 MinPolys_5x5.csv (2MB) MinPolys_5x5.zip (473KB) MinPolys_5x5.tar.gz (473KB)
6x6 MinPolys_6x6.csv (215MB) MinPolys_6x6.zip (41MB) MinPolys_6x6.tar.gz (41MB)

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 2
3x3 729 166 166 729 5
4x4 59,049 3,317 3,317 59,049 12
5x5 14,348,907 133,255 133,255 14,348,907 28
6x6 10,460,353,203 10,872,459     66

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 21 11 10 16
3x3 307 127 180 166
4x4 9,235 3,229 6,006 3,317
5x5 535,375 180,603 354,772 133,255
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 9 18 9 8        
3x3 211 518 0 211 518      
4x4 14,307 44,742 0 0 14,307 44,742    
5x5 2,881,977 11,466,930 0 0 0 2,881,977 11,466,930  
6x6 1,737,616,657 8,722,736,546            

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS   A000079  
2x2 5 2 14
3x3 9 4 326
4x4 17 8 23,162
5x5 33 16 4,900,146
6x6 65 32 3,075,388,478

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     3      
3x3     15      
4x4     163      
5x5     4,389      
6x6     272,867