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Homesymplectic matrix

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# symplectic matrix

A real $2n\times 2n$ matrix $A\in\mathrm{M}_{{2n}}(\mathbb{R})$ is a *symplectic matrix*
if $AJA^{T}=J$, where $A^{T}$ is the transpose of $A$, and
$J\in\mathrm{O}(2n)$ is the orthogonal matrix

$J=\left(\begin{array}[]{cc}\mathbf{0}&\mathbf{I}_{n}\\ -\mathbf{I}_{n}&\mathbf{0}\end{array}\right).$ |

Here $\mathbf{I}_{n}\in\mathrm{M}_{n}(\mathbb{R})$ is the identity $n\times n$ matrix and $\mathbf{0}\in\mathrm{M}_{n}(\mathbb{R})$ is the zero $n\times n$ matrix.

Symplectic matrices satisfy the following properties:

1. The determinant of a symplectic matrix equals one.

2. With standard matrix multiplication, symplectic $2n\times 2n$ matrices form a group denoted by $\mathrm{Sp}(2n)$.

3. Suppose $\Psi=\begin{pmatrix}A&B\\ C&D\end{pmatrix}$, where $A,B,C,D$ are $n\times n$ matrices. Then $\Psi$ is symplectic if and only if

$AD^{T}-BC^{T}=I,\,\,\,\,\,AB^{T}=BA^{T},\,\,\,\,\,CD^{T}=DC^{T}.$ 4. If $X$ and $Y$ are real $n\times n$ matrices, then $U=X+iY$ is unitary if and only if $\begin{pmatrix}X&-Y\\ Y&X\end{pmatrix}$ is symplectic.

## Mathematics Subject Classification

53D05*no label found*

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## Comments

## symplectic matrix and reciprocal eigenvalues

greetings.

I learn from various sources that the eigenvalues of a symplectic matrix occur in reciprocal pairs. However, i also find that there are some matrices that, while their eigenvalues also occur in reciprocal pairs, they are not symplectic. Therefore,I want to know what are the properties of a matrix that has paired reciprocal eigenvalues.

thanks to everyone who answered my question