Prove that the continuoustime LTI system (15.20) is asymptotically stable (or stable, for short) if and only if all the eigenvalues of the A matrix, λ/(A), i = 1, . . . , n, have (strictly) negative real parts. Prove that the discrete-time LTI system (15.28) is stable if and only if all the eigenvalues of the A matrix, λ_i/(A), i = 1, . . . , n, have moduli (strictly) smaller than one.

Use the expression x(t) = e^Atx₀ for the free response of the continuous-time system, and the expression x(k) = A^kx₀ for the free response of the discrete-time system. You may derive your proof under the assumption that A is diagonalizable.