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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 $V$ has an [[orthonormal basis|0S9GGTR0EX]] that consists of [[eigenvectors|0N6960W0BV]] of $T$.\nThe corresponding eigenvalues are real.\n\n## Remarks\n\nIn this basis, the matrix of $T$ is a real diagonal matrix.\nEquivalently, $V$ is the orthogonal direct sum of the eigenspaces of $T$.\nOver $\\C$, the same conclusion holds, with complex eigenvalues, for each [[normal operator|0C8BR8XRTR]].\nThe matrix form is the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].\n\n## Explanations\n\n### L1 human\n\nA self-adjoint map is a pure stretch along perpendicular axes.\nThe axes are the eigenvectors, and the stretch factors are the real eigenvalues.\nEvery symmetric real matrix is diagonal in a suitable orthonormal basis.\n",
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