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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 *normal* when\n$$T T^* = T^* T,$$\nwhere $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.\n\n## Remarks\n\nSelf-adjoint maps, skew-adjoint maps and unitary maps are normal.\n$T$ is normal if and only if $\\norm{T(v)} = \\norm{T^*(v)}$ for all $v \\in V$.\nOver $\\C$, normal maps are exactly the maps with an orthonormal basis of eigenvectors.\n\n## Explanations\n\n### L1 human\n\nA normal map commutes with its adjoint.\nThis condition is weaker than self-adjoint, but it still gives perpendicular eigenvectors over $\\C$.\nRotations of the plane are normal, but not self-adjoint.\n",
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