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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $A$ and $B$ be $n \\times n$ complex [[matrices|0JXZKX526R]].\nThen\n$$e^{A + B} = \\lim_{m \\to \\infty} \\left( e^{A/m} e^{B/m} \\right)^m,$$\nwhere $e^X$ is the [[matrix exponential|020M04FN1H]] and the [[limit|05M5SWSZZD]] is taken entry by entry.\n\n## Remarks\n\nLet $X_m = e^{A/m} e^{B/m}$ and $Y_m = e^{(A + B)/m}$.\nBoth equal $I + (A + B)/m + O(1/m^2)$, so $\\norm{X_m - Y_m} = O(1/m^2)$.\nThe identity $X^m - Y^m = \\sum_{k=0}^{m-1} X^k (X - Y) Y^{m-1-k}$ and the bound $\\norm{X_m}, \\norm{Y_m} \\le e^{(\\norm{A} + \\norm{B})/m}$ give $\\norm{X_m^m - Y_m^m} = O(1/m)$, and $Y_m^m = e^{A + B}$.\nIf $AB = BA$, then $e^{A/m} e^{B/m} = e^{(A + B)/m}$ and the formula holds for every $m$.\nIn general $e^{A + B} \\ne e^A e^B$: for $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $B = \\begin{pmatrix} 0 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix}$, $e^A e^B = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$, while $e^{A + B} = \\begin{pmatrix} \\cosh 1 \u0026 \\sinh 1 \\\\ \\sinh 1 \u0026 \\cosh 1 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nFor numbers, $e^{a + b} = e^a e^b$, but for matrices this fails when $A$ and $B$ do not commute.\nThe formula repairs this: alternate many tiny steps of $A$ and of $B$, and the result approaches $e^{A + B}$.\nIt is like walking diagonally by taking many small steps east and north in turn.\n\n### L2 human\n\nThe formula is the basis of splitting methods: to solve $x' = (A + B)x$, one alternates the flows of $x' = Ax$ and $x' = Bx$, which may be easier to compute.\nIn a Lie group it shows that the sum in the Lie algebra corresponds to a limit of products in the group.\nTrotter extended it to certain unbounded operators on Hilbert spaces, where it is called the Trotter product formula.\nIt is a key step in the proof of the [[Golden–Thompson inequality|Z7R754JC7N]].\n",
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