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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-2-13\">例 2.13</a> \n>\n> 设 $\\mathsf{J}$ 为 $2n\\times2n$ 矩阵\n>\n> $$\n> \\mathsf{J}=\\begin{pmatrix}\\mathsf{O}&\\mathsf{I}\\\\-\\mathsf{I}&\\mathsf{O}\\end{pmatrix},\n> $$\n>\n> 其中 $\\mathsf{O}$ 和 $\\mathsf{I}$ 分别是 $n\\times n$ 零矩阵和单位矩阵。若 $2n\\times2n$ 实矩阵 $\\mathsf{A}$ 满足\n>\n> <span id=\"equation-2-11\" class=\"block-anchor\"></span>\n>\n> $$\n> \\mathsf{A}^{T}\\mathsf{J}\\mathsf{A}=\\mathsf{J},\\tag{2.11}\n> $$\n>\n> 就称它为<u id=\"concept-2-symplectic-matrix\" class=\"concept-term\" data-concept-identity=\"urn:uuid:9452675b-d56a-4a38-90c9-9efa189f3d94\" data-concept-route=\"/concept-identities#concept-9452675b-d56a-4a38-90c9-9efa189f3d94\">辛矩阵(symplectic matrix)</u>。证明这些矩阵组成群的方法与刚才证明 $O(p,q)$ 的方法基本相同。由于 $\\det\\mathsf{J}=1$，<a href=\"#equation-2-11\" class=\"cross-reference\" id=\"xref-03-matrix-groups-13\" data-reference=\"equation-2-11\">式 (2.11)</a> 立即给出 $\\det\\mathsf{A}=\\pm1$，因此 $\\mathsf{A}$ 非奇异。这一群记作 $Sp(2n)$，称为$2n$ 阶<u id=\"concept-2-symplectic-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:459d2b29-b1f3-4298-b1f9-2df232c37b62\" data-concept-route=\"/concept-identities#concept-459d2b29-b1f3-4298-b1f9-2df232c37b62\">辛群(symplectic group)</u>。",
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