Suppose that is matrix where A, B, C and D are&n

Question

Suppose that $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$ is $2n \times 2n$ matrix where A, B, C and D are $n \times n$ matrices. Show that M is symplectic if and only if the following conditions are satisfied:
$\quad$ $A^t D - C^t B = \mathbf{I}$
$\quad$ $A^t C - C^t A = \mathbf{0}$
$\quad$ $B^t D - D^t B = \mathbf{0}$
$\quad$ (Hint: Use block matrix multiplication.)
$\quad$ Also, check that the matrix $\begin{bmatrix} \mathbf{0} & -A \\ A & \mathbf{0} \end{bmatrix}$ , where A is a $n \times n$ orthogonal matrix, is a symplectic matrix.

09 Jan 2026

Answer :

Word Count : 146

Let (J=\begin{bmatrix}0&I\-I&0\end{bmatrix}). By definition, a (2n\times 2n) matrix (M) is symplectic if and only if [ M^{t}JM=J. ] Write [ M=\begin{bmatrix}A&B\ C&D\end{bmatrix},\qquad M^{t}=\begin{bmatrix}A^{t}&C^{t}\ B^{t}&D^{t}\end{bmatrix}. ] First compute [ JM=\begin{bmatrix}0&I\-I&0\end{bmatrix} \begin{bmatrix}A&B\ C&D\end{bmatrix} =\begin{bmatrix}C&D\-A&-B\end{bmatrix}. ] Then [ M^{t}JM= ______ ________ ____ ____ _____ __________ ___ ______ _____ ____.
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