Chapter 9.6, Problem 23E

Stability: In Problem $49$ of Exercises $9.5$, (page $542$), we discussed the notion of stability and asymptotic stability for a linear system of the form $x^{\prime }\left(t\right)=Ax\left(t\right)$. Assume that $A$ has all distinct eigenvalues (real or complex).

a. Show that the system is stable if and only if all the eigenvalues of $A$ have nonpositive real part.

b. Show that the system is asymptotically stable if and only if all the eigenvalues of $A$ have negative real part.