I am trying to find the necessary and sufficient conditions for the point of equilibrium x=0 of $x'=Ax$ to be Lyapunov stable, where A is constant matrix. The book I'm using briefly touches on this, but omits the proof, and I was wondering if anyone could show or point me to a proof that goes though this. I know that the eigenvalues of A have to have negative real parts, and that the proof could be done with Jordan Blocks, but I am not really any good at working with those. Thanks
2026-04-08 19:08:36.1775675316
Proof of Lyapunov Stability for Constant Matrix System
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See the statement and proof in Theorem 14.1 of this lecture note The note also describes in quite detail the Lyapunov stability theory for linear time-invariant system.