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Proofs from Linear ODE systems

All proofs from Linear ODE systems.

[PROOF]

Differentiating the proposed function gives

\mathbf{x}'(t)=\lambda\mathbf{u}e^{\lambda t}.

The equation $\mathbf{x}'=A\mathbf{x}$ therefore becomes

\lambda\mathbf{u}e^{\lambda t} =A\mathbf{u}e^{\lambda t}.

Since $e^{\lambda t}\neq0$ , this is equivalent to $A\mathbf{u}=\lambda\mathbf{u}$ . The reverse implication is the same calculation read backwards.

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.

THExponential modescoupled ODEs: eigenvectors as modesALG 4 / Here