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3347 days ago by jimlb

var('a11,a12,a21,a22,c11,c12,c21,c22,b11,b12,b21,b22,d11,d12,d21,d22') 
       
g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[c11,c12,d11,d12],[c21,c22,d21,d22]]);g; 
       
J=Matrix(SR,[[0,0,1,0],[0,0,0,1],[-1,0,0,0],[0,-1,0,0]]);J; 
       

A general symplectic matrix satisfies $g^{t} J g = J$, so we have:

g*J*g.transpose() 
       
g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[0,c12,d11,d12],[0,0,d21,d22]]);g; 
       
g*J*g.transpose() 
       

We want to restrict to $b_{11} = b_{12} = d_{21}=d_{22} = 0$.

g=Matrix(SR,[[a11,a12,0,0],[a21,a22,b21,b22],[c11,c12,d11,d12],[c21,c22,0,0]]);g; 
       

The condition $g^{t} J g = J$ now reads that the following matrix should be equal to $J$.

g.transpose()*J*g