Hamiltonian matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Hamiltonian matrix is a 2n-by-2n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> A such that JA is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a>, where J is the <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric matrix</a>

J
        =
        
          
            [
            
              
                
                  
                    0
                    
                      n
                    
                  
                
                
                  
                    I
                    
                      n
                    
                  
                
              
              
                
                  −
                  
                    I
                    
                      n
                    
                  
                
                
                  
                    0
                    
                      n
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle J={\begin{bmatrix}0_{n}&I_{n}\\-I_{n}&0_{n}\\\end{bmatrix}}}

and In is the n-by-n <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes the <a href="/facts/Transpose/8wmsagGS">transpose</a>. 
(Not to be confused with <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian (quantum mechanics)</a>)

Hamiltonian matrix open-in-new

Hamiltonian matrix