Question 7.6.1: Consider a dynamical system x(t + 1) = Ax(t), where A is an ......
Consider a dynamical system \vec{x}(t + 1) = A\vec{x}(t), where A is an n × n matrix. Suppose an initial state vector \vec{x}_0 is given. We are told that A has n distinct complex eigenvalues, λ_1,…,λ_n, and that |λ_i| < 1 for i = 1,…, n. What can you say about the long-term behavior of the system, that is, about \lim\limits_{t \to \infty}\vec{x}(t)?
For each complex eigenvalue λ_i , we can choose a complex eigenvector \vec{v}_i . Then the v_i form a complex eigenbasis for A (by Theorem 7.3.4). We can write \vec{x}_0 as a complex linear combination of the \vec{v}_i :
\vec{x}_0 = c_1\vec{v}_1 +···+ c_n\vec{v}_n.
Then
\vec{x}(t) = A^t\vec{x}_0 = c_1λ^t_1\vec{v}_1 +···+ c_nλ^t_n\vec{v}_n.
By Example 5 of Section 7.5,
\lim\limits_{t \to \infty}λ^t_i, since |λ_i| < 1.
Therefore
\lim\limits_{t \to \infty}\vec{x}(t)=\vec{0}.