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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.\nLet $v$ and $w$ be eigenvectors of $T$ for distinct eigenvalues $\\lambda \\ne \\mu$.\nThen $v$ and $w$ are [[orthogonal|0Y2X7P2WB7]].\nSo the [[eigenspaces|0G0RFPG8HR]] of $T$ for distinct eigenvalues are orthogonal to each other.\n\n## Remarks\n\nThe proof: $\\lambda \\langle v, w \\rangle = \\langle T(v), w \\rangle = \\langle v, T(w) \\rangle = \\mu \\langle v, w \\rangle$, since $\\mu$ is real.\nSo $(\\lambda - \\mu) \\langle v, w \\rangle = 0$, and $\\langle v, w \\rangle = 0$.\n\n## Explanations\n\n### L1 human\n\nFor a symmetric matrix, eigenvectors for different eigenvalues are perpendicular.\nSo the eigenspaces meet at right angles.\nThis is the first step towards an orthonormal basis of eigenvectors.\n",
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