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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A$ be a real $n \\times n$ matrix.\nThe homogeneous linear system with constant coefficients given by $A$ is the [[system of linear differential equations|0A6QZ367GX]]\n$$x'(t) = A\\, x(t)$$\nfor an unknown differentiable function $x : \\mathbb{R} \\to \\mathbb{R}^n$.\n\n## Remarks\n\nIts solutions are $x(t) = e^{t A} x(0)$. See [[the matrix exponential as the solution|0YC8JT513J]].\nIf $A v = \\lambda v$, then $x(t) = e^{\\lambda t} v$ is a solution. The [[eigenvalue method|08ZWFDJP3H]] builds all solutions from such solutions.\nExample: for $A = \\begin{pmatrix} 0 \u0026 1 \\\\ -1 \u0026 0 \\end{pmatrix}$, the solutions are rotations: $x(t) = \\left(a \\cos t + b \\sin t, -a \\sin t + b \\cos t\\right)$.\n\n## Explanations\n\n### L1 human\n\nThe rate of change of the vector $x$ is the matrix $A$ applied to $x$, and $A$ does not change with time.\nThis is the vector version of $y' = a y$, whose solutions are $e^{a t} y(0)$.\n",
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