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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $U$ be a complex $n \\times n$ matrix.\nThe matrix $U$ is *unitary* when\n$$U^* U = I_n,$$\nwhere $U^*$ is the [[conjugate transpose|0DXKE10PHF]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].\n\n## Remarks\n\nEquivalently, $U$ is invertible with $U^{-1} = U^*$, and then also $U U^* = I_n$.\nEquivalently, the columns of $U$ form an orthonormal basis of $\\C^n$ for the standard inner product.\nThe eigenvalues of a unitary matrix have absolute value $1$, and $\\abs{\\det U} = 1$.\nA real unitary matrix is an [[orthogonal matrix|0W02QESPJQ]].\n\n## Explanations\n\n### L1 human\n\nUnitary matrices are the complex version of orthogonal matrices.\nThey keep all lengths and inner products in $\\C^n$.\nIn quantum mechanics, they describe how a closed system evolves in time.\n",
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