Symplectic matrix

<p>In mathematics, a symplectic matrix is a 
  
    
      
        2
        n
        ×
        2
        n
      
    
    {\displaystyle 2n\times 2n}
  
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 with <a href="/facts/Real_number/R02gw5Pb">real</a> entries that satisfies the condition
</p>
<table><tbody><tr><td>                              M                      T                          Ω        M        =        Ω        ,              {\displaystyle M^{\text{T}}\Omega M=\Omega ,}  </td> <td></td> <td>1</td></tr></tbody></table>
<p>where 
  
    
      
        
          M
          
            T
          
        
      
    
    {\displaystyle M^{\text{T}}}
  
 denotes the <a href="/facts/Transpose/8wmsagGS">transpose</a> of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 and 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 is a fixed 
  
    
      
        2
        n
        ×
        2
        n
      
    
    {\displaystyle 2n\times 2n}
  
 <a href="/facts/Nonsingular_matrix/fPqXk3V8">nonsingular</a>, <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric matrix</a>. This definition can be extended to 
  
    
      
        2
        n
        ×
        2
        n
      
    
    {\displaystyle 2n\times 2n}
  
 matrices with entries in other <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>, such as the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, <a href="/facts/Finite_field/UkMGJo3m">finite fields</a>, <a href="/facts/P-adic_number/SDgDbkLF"><i>p</i>-adic numbers</a>, and <a href="/facts/Function_field_of_an_algebraic_variety/54cc62wP">function fields</a>.
</p><p>Typically 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 is chosen to be the <a href="/facts/Block_matrix/fPI8HX9f">block matrix</a>

Ω
        =
        
          
            [
            
              
                
                  0
                
                
                  
                    I
                    
                      n
                    
                  
                
              
              
                
                  −
                  
                    I
                    
                      n
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}

where 
  
    
      
        
          I
          
            n
          
        
      
    
    {\displaystyle I_{n}}
  
 is the 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. The matrix 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 has <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 
  
    
      
        +
        1
      
    
    {\displaystyle +1}
  
 and its inverse is 
  
    
      
        
          Ω
          
            −
            1
          
        
        =
        
          Ω
          
            T
          
        
        =
        −
        Ω
      
    
    {\displaystyle \Omega ^{-1}=\Omega ^{\text{T}}=-\Omega }
  
.
</p>

Symplectic matrix open-in-new

Symplectic matrix