(Solution) - Prove Proposition 9 17 A particularly important class of systems are the -(2025 Original AI-Free Solution)

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Prove Proposition 9.17.

A particularly important class of systems are the linear gradient flows
in which AT is a symmetric, positive definite matrix. According to Theorem 8.23, all the eigenvalues of K are real and positive, and so the eigenvalues of the negative definite coefficient matrix - K for the gradient flow system (9.18) are real and negative. Applying Theorem 9.15, we conclude that the zero solution to any gradient flow system (9.18) with negative definite coefficient matrix - K is asymptotically stable.