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==Peano–Baker series==
The most general transition matrix is given by a [[product integral]],
:<math>\begin{align}
\mathbf{\Phi}(t,\tau) = \mathbf{I} &+ \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 \\
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&+ \cdots
\end{align}</math>
where <math>\mathbf{I}</math> is the [[identity matrix]]. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh /> The series has a formal sum that can be written as
:<math>\mathbf{\Phi}(t,\tau) = \exp \mathcal{T}\int_\tau^t\mathbf{A}(\sigma)\,d\sigma</math>
where <math>\mathcal{T}</math> is the [[time-ordering]] operator, used to ensure that the repeated [[product integral]] is in proper order. The [[Magnus expansion]] provides a means for evaluating this product.
==Other properties==
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