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  "body": "\n## Definition\n\nLet $A : [0, \\infty) \\to \\C^{n \\times n}$ be continuous, and let $Y$ be the solution of the [[linear system|0A6QZ367GX]] $Y' = A(t)\\, Y$ with $Y(0) = I$.\nFor matrices $X, Z$ write $[X, Z] = XZ - ZX$.\nThe *Magnus expansion* is the series $\\Omega(t) = \\sum_{k \\ge 1} \\Omega_k(t)$ with $Y(t) = e^{\\Omega(t)}$, the [[matrix exponential|0K665H5V1S]].\nIts first terms are\n$$\\Omega_1(t) = \\int_0^t A(t_1)\\, dt_1, \\qquad \\Omega_2(t) = \\frac12 \\int_0^t \\int_0^{t_1} [A(t_1), A(t_2)]\\, dt_2\\, dt_1,$$\nand in general $\\Omega_k$ is a sum of $k$-fold iterated integrals of nested commutators of $k$ values of $A$.\nThe terms are determined recursively by the differential equation $\\Omega' = \\sum_{j \\ge 0} \\frac{B_j}{j!} \\operatorname{ad}_\\Omega^j(A)$, $\\Omega(0) = 0$, where $B_j$ are the Bernoulli numbers with $B_1 = -\\tfrac12$ and $\\operatorname{ad}_\\Omega(X) = [\\Omega, X]$.\n\n## Remarks\n\nThe series converges, and gives the solution, when $\\int_0^t \\norm{A(s)}_2\\, ds \u003c \\pi$, where $\\norm{\\cdot}_2$ is the spectral norm; this is a sufficient condition.\nExample: if $A(t_1)$ and $A(t_2)$ commute for all $t_1, t_2$, all terms with $k \\ge 2$ vanish and $Y(t) = \\exp\\left(\\int_0^t A\\right)$; for $A(t) = a(t) N$ with a fixed matrix $N$ and scalar $a$, this gives $Y = e^{(\\int_0^t a) N}$.\nWithout commutation, $\\exp\\left(\\int_0^t A\\right)$ is in general not the solution, and $\\Omega_2$ is the first correction.\nThe reason is that the derivative of $\\exp(\\Omega_1(t))$ need not be $A(t) \\exp(\\Omega_1(t))$ when $A(t)$ does not commute with $\\Omega_1(t)$.\nLet $A(t)$ lie in a matrix Lie algebra for all $t$, such as the skew-symmetric matrices.\nThen every truncation of the series also lies in it, so its exponential lies in the corresponding group.\n",
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