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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ complex matrix.\nThe matrix $A$ is *normal* when\n$$A A^* = A^* A,$$\nwhere $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.\n\n## Remarks\n\nHermitian, skew-Hermitian and unitary matrices are normal.\nA real matrix is normal when $A A^T = A^T A$; real symmetric and orthogonal matrices are examples.\nThe matrix $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not normal.\nA matrix is normal if and only if its map on $\\C^n$ is a [[normal operator|0C8BR8XRTR]] for the standard inner product.\n\n## Explanations\n\n### L1 human\n\nA matrix is normal when it commutes with its conjugate transpose.\nNormal matrices are exactly the matrices that a unitary change of basis makes diagonal.\nThey include the symmetric, the Hermitian and the unitary matrices.\n",
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