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  "body": "\n## Definition\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$, and let $T \\colon V \\to V$ be an [[endomorphism|0J7H0YBFDJ]].\nThe map $T$ is *self-adjoint* when $T^* = T$, where $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.\nEquivalently, $\\langle T(v), w \\rangle = \\langle v, T(w) \\rangle$ for all $v, w \\in V$.\n\n## Remarks\n\nIn an orthonormal basis, $T$ is self-adjoint if and only if its matrix is [[symmetric|0VR671PHE5]] (over $\\R$) or [[Hermitian|0P17NGQQMQ]] (over $\\C$).\nOrthogonal projections are self-adjoint.\nThe eigenvalues of a self-adjoint map are real, see [[the reality of the eigenvalues|0C4NT4GT3X]].\nSelf-adjoint maps are the subject of the [[spectral theorem|06J4PXRG22]].\n\n## Explanations\n\n### L1 human\n\nA self-adjoint map can move freely from one side of the inner product to the other.\nFor real matrices, these are the symmetric matrices.\nThey model observable quantities in quantum mechanics, because their eigenvalues are real.\n",
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