# No name

## 3347 days ago by jimlb

var('a11,a12,a21,a22,c11,c12,c21,c22,b11,b12,b21,b22,d11,d12,d21,d22')
 $\newcommand{\Bold}{\mathbf{#1}}\left(a_{11}, a_{12}, a_{21}, a_{22}, c_{11}, c_{12}, c_{21}, c_{22}, b_{11}, b_{12}, b_{21}, b_{22}, d_{11}, d_{12}, d_{21}, d_{22}\right)$
g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[c11,c12,d11,d12],[c21,c22,d21,d22]]);g;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} a_{11} & a_{12} & b_{11} & b_{12} \\ a_{21} & a_{22} & b_{21} & b_{22} \\ c_{11} & c_{12} & d_{11} & d_{12} \\ c_{21} & c_{22} & d_{21} & d_{22} \end{array}\right)$
J=Matrix(SR,[[0,0,1,0],[0,0,0,1],[-1,0,0,0],[0,-1,0,0]]);J;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{array}\right)$

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

g*J*g.transpose()
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & a_{11} b_{21} + a_{12} b_{22} - a_{21} b_{11} - a_{22} b_{12} & a_{11} d_{11} + a_{12} d_{12} - b_{11} c_{11} - b_{12} c_{12} & a_{11} d_{21} + a_{12} d_{22} - b_{11} c_{21} - b_{12} c_{22} \\ -a_{11} b_{21} - a_{12} b_{22} + a_{21} b_{11} + a_{22} b_{12} & 0 & a_{21} d_{11} + a_{22} d_{12} - b_{21} c_{11} - b_{22} c_{12} & a_{21} d_{21} + a_{22} d_{22} - b_{21} c_{21} - b_{22} c_{22} \\ -a_{11} d_{11} - a_{12} d_{12} + b_{11} c_{11} + b_{12} c_{12} & -a_{21} d_{11} - a_{22} d_{12} + b_{21} c_{11} + b_{22} c_{12} & 0 & c_{11} d_{21} + c_{12} d_{22} - c_{21} d_{11} - c_{22} d_{12} \\ -a_{11} d_{21} - a_{12} d_{22} + b_{11} c_{21} + b_{12} c_{22} & -a_{21} d_{21} - a_{22} d_{22} + b_{21} c_{21} + b_{22} c_{22} & -c_{11} d_{21} - c_{12} d_{22} + c_{21} d_{11} + c_{22} d_{12} & 0 \end{array}\right)$
g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[0,c12,d11,d12],[0,0,d21,d22]]);g;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} a_{11} & a_{12} & b_{11} & b_{12} \\ a_{21} & a_{22} & b_{21} & b_{22} \\ 0 & c_{12} & d_{11} & d_{12} \\ 0 & 0 & d_{21} & d_{22} \end{array}\right)$
g*J*g.transpose()
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & a_{11} b_{21} + a_{12} b_{22} - a_{21} b_{11} - a_{22} b_{12} & a_{11} d_{11} + a_{12} d_{12} - b_{12} c_{12} & a_{11} d_{21} + a_{12} d_{22} \\ -a_{11} b_{21} - a_{12} b_{22} + a_{21} b_{11} + a_{22} b_{12} & 0 & a_{21} d_{11} + a_{22} d_{12} - b_{22} c_{12} & a_{21} d_{21} + a_{22} d_{22} \\ -a_{11} d_{11} - a_{12} d_{12} + b_{12} c_{12} & -a_{21} d_{11} - a_{22} d_{12} + b_{22} c_{12} & 0 & c_{12} d_{22} \\ -a_{11} d_{21} - a_{12} d_{22} & -a_{21} d_{21} - a_{22} d_{22} & -c_{12} d_{22} & 0 \end{array}\right)$

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;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & b_{21} & b_{22} \\ c_{11} & c_{12} & d_{11} & d_{12} \\ c_{21} & c_{22} & 0 & 0 \end{array}\right)$

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

g.transpose()*J*g
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & -a_{11} c_{12} + a_{12} c_{11} - a_{21} c_{22} + a_{22} c_{21} & -a_{11} d_{11} + b_{21} c_{21} & -a_{11} d_{12} + b_{22} c_{21} \\ a_{11} c_{12} - a_{12} c_{11} + a_{21} c_{22} - a_{22} c_{21} & 0 & -a_{12} d_{11} + b_{21} c_{22} & -a_{12} d_{12} + b_{22} c_{22} \\ a_{11} d_{11} - b_{21} c_{21} & a_{12} d_{11} - b_{21} c_{22} & 0 & 0 \\ a_{11} d_{12} - b_{22} c_{21} & a_{12} d_{12} - b_{22} c_{22} & 0 & 0 \end{array}\right)$