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  "body": "\n## Definition\n\nLet $I$ be an open interval and let $A : I \\to \\mathbb{R}^{n \\times n}$ be continuous.\nA fundamental matrix of the homogeneous [[linear system|0A6QZ367GX]] $x' = A(t) x$ is a differentiable function $\\Phi : I \\to \\mathbb{R}^{n \\times n}$ with these two properties.\nFor each $t \\in I$, $\\Phi'(t) = A(t)\\, \\Phi(t)$.\nFor each $t \\in I$, the matrix $\\Phi(t)$ is invertible.\n\n## Remarks\n\nEquivalently, the columns of $\\Phi$ are $n$ solutions that form a basis of the space of solutions.\nEach solution is $x(t) = \\Phi(t)\\, c$ for a unique $c \\in \\mathbb{R}^n$, and the solution with $x(t_0) = x_0$ is $\\Phi(t) \\Phi(t_0)^{-1} x_0$.\nIf $\\Phi'(t) = A(t) \\Phi(t)$ and $\\Phi(t_0)$ is invertible at one point $t_0$, then $\\Phi(t)$ is invertible at each point, by [[Liouville's formula|09PS3S0EF6]].\nFor a constant matrix $A$, $\\Phi(t) = e^{tA}$ is a fundamental matrix. See [[the matrix exponential as the solution|0YC8JT513J]].\n\n## Explanations\n\n### L1 human\n\nA fundamental matrix puts $n$ independent solutions side by side as columns.\nThen one matrix product gives each solution: choose the vector $c$ of coefficients.\n",
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