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  "body": "\n## Summary\n\nThe Magnus expansion writes the solution of a linear system $Y' = A(t)\\, Y$, $Y(0) = I$, as one [[matrix exponential|020M04FN1H]] $Y(t) = e^{\\Omega(t)}$.\nThe exponent $\\Omega(t)$ is a series of iterated integrals of commutators of $A$ at different times.\nWhen the matrices $A(t)$ commute with each other, only the first term $\\int_0^t A$ remains.\nTruncations of the series keep structure, for example they stay in a matrix group, which makes them useful in numerical methods and quantum mechanics.\nSee the [[definition|TD9T02HKPD]].\n\n## Explanations\n\n### L1 human\n\nFor a scalar equation $y' = a(t)\\, y$ the solution is $y = e^{\\int a}$.\nFor matrices this formula is wrong in general, because matrices at different times need not commute.\nThe Magnus expansion corrects the exponent by extra terms built from commutators $[A(t_1), A(t_2)] = A(t_1) A(t_2) - A(t_2) A(t_1)$.\nThese terms measure how much the matrices fail to commute.\n",
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