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  "body": "\n## Description\n\nLet $\\mathbb{K}$ be a [[field|04C5ZBG1WT]], let $n \\ge 1$, and let\n$$\\Omega = \\begin{pmatrix} 0 \u0026 I_n \\\\ -I_n \u0026 0 \\end{pmatrix},$$\na $2n \\times 2n$ [[block matrix|0ASDFEQS6P]], where $I_n$ is the $n \\times n$ identity matrix.\nThe *symplectic group* is\n$$\\operatorname{Sp}(2n, \\mathbb{K}) = \\set{M \\in \\operatorname{GL}(2n, \\mathbb{K}) : M^T \\Omega M = \\Omega}.$$\nIt is a [[group|G5S1M9]] under matrix multiplication, a subgroup of the [[general linear group|0FCHE902D7]].\nCheck: $I^T \\Omega I = \\Omega$; if $M^T \\Omega M = \\Omega$ and $N^T \\Omega N = \\Omega$, then $(MN)^T \\Omega (MN) = N^T (M^T \\Omega M) N = \\Omega$; and multiplying $M^T \\Omega M = \\Omega$ by $(M^T)^{-1}$ on the left and $M^{-1}$ on the right gives $(M^{-1})^T \\Omega M^{-1} = \\Omega$.\n\n## Remarks\n\nThe group $\\operatorname{Sp}(2n, \\mathbb{K})$ is the group of linear maps of $\\mathbb{K}^{2n}$ that preserve the standard symplectic form $\\omega(x, y) = x^T \\Omega y$; see [[symplectic vector space|PQ7JJY0VPH]].\nEvery symplectic matrix has determinant $1$, so $\\operatorname{Sp}(2n, \\mathbb{K})$ lies in the [[special linear group|0V9PMN0RBH]].\nFor $n = 1$, every $2 \\times 2$ matrix $M$ satisfies $M^T \\Omega M = (\\det M)\\, \\Omega$, so $\\operatorname{Sp}(2, \\mathbb{K}) = \\operatorname{SL}(2, \\mathbb{K})$.\nFor $\\mathbb{K} = \\R$ it is a connected noncompact Lie group of dimension $n(2n + 1)$.\nThe notation $\\operatorname{Sp}(n)$ is also used for the compact symplectic group, the quaternionic unitary group, which is a different group.\n\n## Explanations\n\n### L1 human\n\nSymplectic matrices are the linear changes of coordinates that preserve a certain signed area between pairs of position and momentum coordinates.\nThey are the linear maps allowed in Hamiltonian mechanics, because they keep the form of Hamilton's equations.\nIn the plane, preserving signed area is the same as having determinant $1$.\nIn higher dimensions the condition is stronger than determinant $1$.\n",
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