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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-4-1\">习题 4.1</a> \n>\n> 一个 $n \\times n$ 矩阵 $\\mathsf { T } = [ T _ { j } ^ { i } ]$ 的 <u id=\"concept-4-trace\" class=\"concept-term\" data-concept-identity=\"urn:uuid:761461ce-048a-4c1f-8316-6b32c6043b81\" data-concept-route=\"/concept-identities#concept-761461ce-048a-4c1f-8316-6b32c6043b81\">迹(trace)</u> 定义为其对角元素之和，\n>\n> $$\n> \\operatorname{tr} \\mathsf {T} = T _ {i} ^ {i} = T _ {1} ^ {1} + T _ {2} ^ {2} + \\dots + T _ {n} ^ {n}.\n> $$\n>\n> 证明\n>\n> (a) $\\operatorname { t r } ( \\mathsf { S } \\mathsf { T } ) = \\operatorname { t r } ( \\mathsf { T } \\mathsf { S } )$\n>\n> (b) $\\operatorname { t r } ( \\mathsf { A T A } ^ { - 1 } ) = \\operatorname { t r } T .$\n>\n> (c) 若 $T : V \\to V$ 是任意算符，则定义其迹为它关于基 $\\{ e _ { i } \\}$ 的矩阵的迹。证明此定义与基的选择无关，因此写 $\\operatorname{tr}$ $T$ 时没有歧义。\n>\n> (d) 若 $f ( z ) = a _ { 0 } + a _ { 1 } z + a _ { 2 } z ^ { 2 } + \\cdot \\cdot \\cdot + ( - 1 ) ^ { n } z ^ { n }$ 是算符 $T$ 的特征多项式，证明 $\\operatorname{tr}$ $T = ( - 1 ) ^ { n - 1 } a _ { n - 1 } .$\n>\n> (e) 若 $T$ 具有特征值 $\\lambda _ { 1 } , \\ldots , \\lambda _ { m }$，其重数为 $p _ { 1 } , \\ldots , p _ { m }$，证明\n>\n> $$\n> \\operatorname{tr} T = \\sum_ {i = 1} ^ {m} p _ {i} \\lambda_ {i}.\n> $$",
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