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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $A$ be an $n \\times n$ matrix over $\\mathbb{K}$.\nThe *matrix exponential* of $A$ is the $n \\times n$ matrix\n$$e^A = \\exp(A) = \\sum_{k=0}^{\\infty} \\frac{1}{k!} A^k = I_n + A + \\frac{1}{2} A^2 + \\frac{1}{6} A^3 + \\cdots,$$\nwith $A^0 = I_n$.\n\n## Remarks\n\nThe series converges for each matrix $A$, entry by entry.\nFor a diagonal matrix $D$ with entries $d_i$, $e^D$ is diagonal with entries $e^{d_i}$.\nIf $A = P D P^{-1}$, then $e^A = P e^D P^{-1}$.\nThe function $x(t) = e^{tA} x_0$ solves the linear differential equation $x' = Ax$ with $x(0) = x_0$.\n\n## Explanations\n\n### L1 human\n\nThe matrix exponential uses the familiar series of $e^x$, with a matrix in place of the number $x$.\nIt is the main tool for solving systems of linear differential equations.\nFor $A = \\begin{pmatrix} 0 \u0026 -t \\\\ t \u0026 0 \\end{pmatrix}$, $e^A$ is the rotation of the plane by the angle $t$.\n",
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