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  "body": "\n## Description\n\nLet $n \\ge 1$.\nThe *unitary group* is the set\n$$U(n) = \\{U \\in M_n(\\C) : U^* U = I_n\\}$$\nof complex $n \\times n$ [[unitary matrices|0AR68JTMDT]], with matrix multiplication.\nIt is a subgroup of the general linear group $GL_n(\\C)$.\n\n## Remarks\n\nIt is the group of linear isometries of $\\C^n$ with the standard inner product.\n$U(1)$ is the circle group $\\{z \\in \\C : \\abs{z} = 1\\}$.\nThe unitary matrices with determinant $1$ form the special unitary group $SU(n)$.\n$U(n)$ is a compact connected Lie group of real dimension $n^2$.\n\n## Explanations\n\n### L1 human\n\nThe unitary group collects all complex matrices that keep the length of every vector.\nFor $n = 1$, these are multiplications by complex numbers of absolute value $1$, that is, rotations of the complex plane.\nThe group appears throughout physics.\n",
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