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"quote": "> [!example] 例 2.15 \n>\n> 复矩阵 $\\mathsf{A}$ 的伴随矩阵(adjoint)（共轭转置）定义为 $\\mathsf{A}^{\\dagger}=\\overline{\\mathsf{A}^{T}}$，分量为 $a_{ij}^{\\dagger}=\\overline{a_{ji}}$；复数上的横线表示复共轭。若 $n\\times n$ 复矩阵 $\\mathsf{U}$ 满足\n>\n> \n>\n> $$\n> \\mathsf{U}\\mathsf{U}^{\\dagger}=\\mathsf{I},\\tag{2.12}\n> $$\n>\n> 则称它为酉矩阵(unitary matrix)。立即可得\n>\n> $$\n> \\det\\mathsf{U}\\,\\overline{\\det\\mathsf{U}}=|\\det\\mathsf{U}|^{2}=1.\n> $$\n>\n> 因而存在实数 $\\phi\\in[0,2\\pi)$，使 $\\det\\mathsf{U}=e^{i\\phi}$。所有酉矩阵都非奇异，群性质也容易验证。全体 $n\\times n$ 酉矩阵组成$n$ 阶酉群(unitary group)，记作 $U(n)$。其中行列式为 $1$ 的酉矩阵组成$n$ 阶特殊酉群(special unitary group)，记作 $SU(n)$。",
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"quote": "\n\n> [!definition] Definition · Unitary group\n> \n>\n> Definition (Unitary group). The unitary group $U ( n )$ in $n$ dimensions is the group of invertible transformations preserving a Hermitian inner product on a complex $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ complex invertible matrices satisfying\n>\n> \n>\n> $$\n> L ^ {- 1} = \\overline {{L}} ^ {T} = L ^ {\\dagger}\n> $$\n",
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"quote": "\n\n> [!definition] Definition · Unitary group\n> \n>\n> Definition (Unitary group). The unitary group $U ( n )$ in $n$ dimensions is the group of invertible transformations preserving a Hermitian inner product on a complex $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ complex invertible matrices satisfying\n>\n> \n>\n> $$\n> L ^ {- 1} = \\overline {{L}} ^ {T} = L ^ {\\dagger}\n> $$\n",
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"quote": "我们最感兴趣的是保持内积的线性变换群：[正交群(orthogonal group)](../../../../../concepts/linear-algebra/orthogonal-group)与[酉群(unitary group)](../../../../../concepts/linear-algebra/unitary-group)。我们已经看到，正交群是 ${ \\cal G L } ( n , { \\bf R } )$ 的子群，酉群是 $G L ( n , \\mathbf { C } )$ 的子群；它们的元素 $\\Omega$ 均满足下式",
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"quote": "The groups we are most interested in are the groups of linear transformations preserving an inner product: the [orthogonal group](../../../../../concepts/linear-algebra/orthogonal-group) and [unitary group](../../../../../concepts/linear-algebra/unitary-group). We have seen that these are subgroups of ${ \\cal G L } ( n , { \\bf R } )$ or $G L ( n , \\mathbf { C } )$ , consisting of those elements $\\Omega$ satisfying the condition",
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"quote": "对于[酉群(unitary group)](../../../../../concepts/linear-algebra/unitary-group) $U(n)$，酉性条件要求 $X$ 是[斜伴随矩阵(skew-adjoint matrix)](../../../../../concepts/linear-algebra/skew-adjoint-matrix)（也称为[斜厄米矩阵(skew-Hermitian matrix)](../../../../../concepts/linear-algebra/skew-adjoint-matrix)），并满足",
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"quote": "- [酉群(unitary group)](../../../../../concepts/linear-algebra/unitary-group) $U(n)\\subset GL(n,\\mathbf{C})$ 由满足 $\\Omega^\\dagger=\\Omega^{-1}$ 的 $n\\times n$ 复矩阵 $\\Omega$ 构成。其李代数 $\\mathfrak{u}(n)$ 由满足 $X^\\dagger=-X$ 的 $n\\times n$ 斜厄米矩阵构成。",
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