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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $m, n \\ge 1$, and let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $\\mathbb{K}$.\nThe *Frobenius norm* of $A$ is\n$$\\norm{A}_F = \\sqrt{\\sum_{i=1}^{m} \\sum_{j=1}^{n} \\abs{a_{ij}}^2}.$$\n\n## Remarks\n\nEquivalently, $\\norm{A}_F = \\sqrt{\\tr(A^* A)}$, where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] and $\\tr$ is the [[trace|0XC1RDBY7A]].\nIf $\\sigma_1, \\ldots, \\sigma_r$ are the nonzero [[singular values|09VMDSRXWJ]] of $A$, then $\\norm{A}_F^2 = \\sigma_1^2 + \\cdots + \\sigma_r^2$.\nExample: for $A = \\begin{pmatrix} 1 \u0026 2 \\\\ 3 \u0026 4 \\end{pmatrix}$, $\\norm{A}_F = \\sqrt{30}$.\nThe Frobenius norm is a submultiplicative [[matrix norm|54T6Y6DSY4]], by the Cauchy–Schwarz inequality, and $\\norm{UAV}_F = \\norm{A}_F$ for unitary $U$ and $V$.\nIt is not an operator norm when $n \\ge 2$: every operator norm gives $\\norm{I_n} = 1$, but $\\norm{I_n}_F = \\sqrt{n}$.\n",
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