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where <math>\mathbf{U}(t)</math> is the fundamental solution matrix that satisfies
: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math>
is a <math>n \times n</math> matrix that is a linear mapping onto itself, i.e., with <math>\mathbf{u}(t)=0</math>, given the state <math>\mathbf{x}(\tau)</math> at any time <math>\tau</math>, the state at any other time <math>t</math> is given by the
:<math>\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)</math>
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