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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-5-12\">习题 5.12</a> \n>\n> 若 $V$ 是内积空间， 一个算符$K : V \\to V$称为<u id=\"concept-5-self-adjoint-operator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:c7a78756-912d-4e4f-a31b-762d35ae477b\" data-concept-route=\"/concept-identities#concept-c7a78756-912d-4e4f-a31b-762d35ae477b\">自伴算符(self-adjoint operator)</u>，如果\n>\n> $$\n> \\langle u | K v \\rangle = \\langle K u | v \\rangle\n> $$\n>\n> 对任意一对向量 $u$、$v \\in V$。设 $\\{ e _ { i } \\}$ 为任意基，具有 $\\langle e _ { i } \\mid e _ { j } \\rangle = h _ { i j }$，并令 $K e _ { k } =$ $K _ { k } ^ { j } e _ { j }$。证明：若 $\\mathsf { H } = [ h _ { i j } ]$ 且 $\\mathsf { K } = [ K _ { j } ^ { k } ]$，则\n>\n> $$\n> \\mathsf {H K} = \\mathsf {K} ^ {\\dagger} \\mathsf {H} = (\\mathsf {H K}) ^ {\\dagger}.\n> $$\n>\n> 若 $\\left\\{ e _ { i } \\right\\}$ 是一组标准正交基，证明 $K$ 是埃尔米特矩阵。",
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