Bohemian Matrices

UH_1_Subdiag_n1_0

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

Characteristic Polynomials

Download all characteristic polynomials:

Matrix Size .csv .zip .tar.gz
2x2 CharPolys_2x2.csv (66B) CharPolys_2x2.zip (224B) CharPolys_2x2.tar.gz (171B)
3x3 CharPolys_3x3.csv (302B) CharPolys_3x3.zip (294B) CharPolys_3x3.tar.gz (241B)
4x4 CharPolys_4x4.csv (2KB) CharPolys_4x4.zip (789B) CharPolys_4x4.tar.gz (763B)
5x5 CharPolys_5x5.csv (34KB) CharPolys_5x5.zip (8KB) CharPolys_5x5.tar.gz (8KB)
6x6 CharPolys_6x6.csv (777KB) CharPolys_6x6.zip (154KB) CharPolys_6x6.tar.gz (154KB)
7x7 CharPolys_7x7.csv (29MB) CharPolys_7x7.zip (5MB) CharPolys_7x7.tar.gz (5MB)
8x8 CharPolys_8x8.csv (2GB) CharPolys_8x8.zip (283MB) CharPolys_8x8.tar.gz (283MB)

Minimal Polynomials

Download all minimal polynomials:

Matrix Size .csv .zip .tar.gz
2x2 MinPolys_2x2.csv (66B) MinPolys_2x2.zip (222B) MinPolys_2x2.tar.gz (168B)
3x3 MinPolys_3x3.csv (302B) MinPolys_3x3.zip (292B) MinPolys_3x3.tar.gz (238B)
4x4 MinPolys_4x4.csv (2KB) MinPolys_4x4.zip (787B) MinPolys_4x4.tar.gz (761B)
5x5 MinPolys_5x5.csv (34KB) MinPolys_5x5.zip (8KB) MinPolys_5x5.tar.gz (8KB)
6x6 MinPolys_6x6.csv (777KB) MinPolys_6x6.zip (154KB) MinPolys_6x6.tar.gz (154KB)
7x7 MinPolys_7x7.csv (29MB) MinPolys_7x7.zip (5MB) MinPolys_7x7.tar.gz (5MB)
8x8 MinPolys_8x8.csv (2GB) MinPolys_8x8.zip (283MB) MinPolys_8x8.tar.gz (283MB)

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 28 28 64 5
4x4 1,024 198 198 1,024 12
5x5 32,768 2,291 2,291 32,768 28
6x6 2,097,152 42,900 42,900 2,097,152 66
7x7 268,435,456 1,336,212     168
8x8 68,719,476,736 68,737,820     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 8 2 6 6
3x3 50 14 36 28
4x4 515 51 464 198
5x5 8,192 1,478 6,714 2,291
6x6 209,340 23,538 185,802 42,900
7x7 8,227,106 1,208,206 7,018,900  
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 3 5 3 5            
3x3 17 47 0 17 47          
4x4 183 841 0 0 183 841        
5x5 3,769 28,999 0 0 0 3,769 28,999      
6x6 149,607 1,947,545 0 0 0 0 149,607 1,947,545    
7x7 11,522,393 256,913,063                
8x8 1,731,779,367 66,987,697,369                

Determinant

Matrix Size Number of Distinct Determinants Maximum Determinant Number of Unimodular Matrices
OEIS   A000079  
2x2 3 2 4
3x3 5 4 28
4x4 9 8 352
5x5 17 16 8,128
6x6 33 32 351,232
7x7 65 64 28,831,744
8x8 129 128 4,548,591,616

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      
8x8     1