{
  "aliases": [
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  "body": "\n## Statement\n\nLet $I$ be an open interval, let $A : I \\to \\mathbb{R}^{n \\times n}$ be continuous, and let $\\Phi : I \\to \\mathbb{R}^{n \\times n}$ be a differentiable function with $\\Phi'(t) = A(t)\\, \\Phi(t)$ for each $t \\in I$.\nThen for each $t_0, t \\in I$,\n$$\\det \\Phi(t) = \\det \\Phi(t_0)\\, \\exp\\left( \\int_{t_0}^{t} \\operatorname{tr} A(s)\\, ds \\right),$$\nwhere $\\operatorname{tr} A(s)$ is the trace of $A(s)$.\n\n## Remarks\n\nSo $\\det \\Phi(t)$ is zero at each point of $I$ or at no point of $I$. This shows that a [[fundamental matrix|0MQ24RBG93]] is invertible everywhere if it is invertible at one point.\nFor a constant matrix $A$, the formula gives $\\det e^{tA} = e^{t \\operatorname{tr} A}$.\nFor the system that comes from $y'' + p y' + q y = 0$, the trace is $-p$, and the formula is [[Abel's identity|0VGWKX2JHB]].\n\n## Explanations\n\n### L1 human\n\nThe determinant of $\\Phi$ is the volume spanned by $n$ solutions.\nThe formula says how this volume changes in time: its rate of growth is the trace of $A$.\n",
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    "name": "Liouville's formula",
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        "locator": "Statement of Liouville's formula",
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  "name": "Liouville's formula",
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