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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ complex matrix.\nThen $A$ is [[normal|0MR0BYF0K2]] if and only if there are a [[unitary matrix|0AR68JTMDT]] $U$ and a diagonal matrix $D$ with\n$$A = U D U^*.$$\n\n## Remarks\n\nThe columns of $U$ are then an orthonormal basis of $\\C^n$ of eigenvectors of $A$.\nIf $A$ is Hermitian, then $D$ is real; if $A$ is unitary, then the diagonal entries of $D$ have absolute value $1$.\nOver $\\R$, a normal matrix need not be diagonalisable: a rotation of the plane is normal.\n\n## Explanations\n\n### L1 human\n\nThe normal matrices are exactly the complex matrices that are diagonal in some orthonormal basis.\nThis generalises the spectral theorem from Hermitian matrices to all normal matrices.\nThe eigenvalues can now be any complex numbers.\n",
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