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  "body": "\n## Statement\n\nLet $A$ be a [[Hermitian|0P17NGQQMQ]] $n \\times n$ complex matrix.\nThen there are a [[unitary matrix|0AR68JTMDT]] $U$ and a real diagonal matrix $D$ with\n$$A = U D U^*.$$\nIf $A$ is a real [[symmetric matrix|0VR671PHE5]], then $U$ can be chosen real, that is, [[orthogonal|0W02QESPJQ]]: $A = Q D Q^T$.\n\n## Remarks\n\nThe diagonal entries of $D$ are the eigenvalues of $A$, and the columns of $U$ are orthonormal eigenvectors.\nThis is the matrix form of the [[spectral theorem|06J4PXRG22]].\nFor example, $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix} = Q \\begin{pmatrix} 3 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix} Q^T$ with $Q = \\frac{1}{\\sqrt 2} \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 -1 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nEvery Hermitian matrix is diagonal after a unitary change of coordinates.\nThe new coordinate axes are perpendicular eigenvectors.\nFor real symmetric matrices, this is a rotation, possibly with a reflection, of the axes.\n",
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