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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$, and let $T \\colon V \\to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.\nThen each [[eigenvalue|0BQAA2BK3Q]] of $T$ is a real number.\n\n## Remarks\n\nIf $T(v) = \\lambda v$ with $v \\ne 0$, then $\\lambda \\langle v, v \\rangle = \\langle T(v), v \\rangle = \\langle v, T(v) \\rangle = \\bar\\lambda \\langle v, v \\rangle$, so $\\lambda = \\bar\\lambda$.\nIn particular, a [[Hermitian matrix|0P17NGQQMQ]] has only real eigenvalues.\nSo a real [[symmetric matrix|0VR671PHE5]] has only real eigenvalues, also when we view it as a complex matrix.\n\n## Explanations\n\n### L1 human\n\nSelf-adjoint maps never have complex eigenvalues that are not real.\nFor example, the symmetric matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ has the eigenvalues $1$ and $-1$.\nBy contrast, the rotation matrix $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is not symmetric, and its eigenvalues are $\\pm i$.\n",
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