Mr. Integral
Math notes/Differential Equations/Systems and Phase Planes

Matrix Exponentials and Repeated Modes — Cheat sheet

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Core rule

x(t)=eAtx0,\mathbf x(t)=e^{At}\mathbf x_0, e(λI+N)t=eλt∑k≥0tkNkk!.e^{(\lambda I+N)t}=e^{\lambda t}\sum_{k\ge0}\frac{t^kN^k}{k!}.

Watch for

A negative repeated eigenvalue can produce a transient polynomial factor before eventual decay. A plot over a short interval may show growth in one component even though all states eventually approach zero.