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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be a connected open set, and let $A \\colon U \\to \\C^{n \\times n}$ be [[holomorphic|0NAHXR1FRH]].\nConsider the [[linear system|037ZNY7NAG]] $Y' = A(z)\\, Y$, fix $z_0 \\in U$, and let $\\Phi$ be a [[fundamental matrix|0H1GN6P9R5]] of the system on a disc around $z_0$ in $U$.\nLet $\\gamma$ be a loop in $U$ based at $z_0$.\nThe [[analytic continuation|0M3YQJ81NB]] of $\\Phi$ along $\\gamma$ is again a fundamental matrix near $z_0$, so it equals $\\Phi\\, M_\\gamma$ for a unique matrix $M_\\gamma \\in \\mathrm{GL}_n(\\C)$.\nThe matrix $M_\\gamma$ is the *monodromy matrix* of $\\gamma$.\nIt depends only on the homotopy class $[\\gamma]$ of $\\gamma$ in the fundamental group $\\pi_1(U, z_0)$.\nThe map\n$$\\rho \\colon \\pi_1(U, z_0) \\to \\mathrm{GL}_n(\\C), \\qquad \\rho([\\gamma]) = M_\\gamma^{-1},$$\nis a group homomorphism, the *monodromy representation* of the system, and its image is the *monodromy group*.\n\n## Remarks\n\nContinuing $\\Phi$ along $\\gamma_1$ and then along $\\gamma_2$ gives $\\Phi M_{\\gamma_2} M_{\\gamma_1}$, so $\\gamma \\mapsto M_\\gamma$ reverses products; taking inverses makes $\\rho$ a [[representation|WYGRXCG2RN]] of $\\pi_1$ in the [[general linear group|0FCHE902D7]].\nChanging $\\Phi$ to $\\Phi C$ with $C$ invertible conjugates every $M_\\gamma$ by $C$, so the representation is determined up to conjugation.\nThe same definition applies to a scalar equation of order $n$, using a fundamental system of solutions.\n\nExample: for $y' = \\frac{a}{z} y$ on $U = \\C \\setminus \\set{0}$ with $a \\in \\C$, take $\\Phi(z) = z^a = e^{a \\log z}$ near $z_0 = 1$.\nContinuation once counterclockwise around $0$ adds $2 \\pi i$ to $\\log z$, so $M_\\gamma = e^{2 \\pi i a}$.\nThe monodromy is trivial exactly when $a \\in \\Z$.\n\nNon-example: for $y' = y$ on $\\C$, the monodromy is trivial.\nThe solution $e^z$ is single-valued and $U = \\C$ is simply connected, so every monodromy matrix is $1$.\n",
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