Definition:Matrix Exponential

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The matrix exponential, denoted $e^{t \mathbf A}$, of the order $n$ constant square matrix $\mathbf A$, is defined to be the unique solution to the initial value problem:

$X' = \mathbf A X, \ \map X {\mathbf 0} = \mathbf I$


$\mathbf I$ is the unit matrix
$X$ is an order $n$ square matrix which is a function of $t$.