Question

Let \(A\) be a \(3 \times 3\) matrix with eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) and corresponding linearly independent eigenvectors \(V_{1}, V_{2}, V_{3}\) which we can therefore use as a basis.

a) If \(X=a V_{1}+b V_{2}+c V_{3}\), compute \(A X, A^{2} X\), and \(A^{35} X\) in terms of \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), \(V_{1}, V_{2}, V_{3}, a, b\) and \(c\) (only).
b) If \(\lambda_{1}=1,\left|\lambda_{2}\right|<1\), and \(\left|\lambda_{3}\right|<1\), compute \(\lim _{k \rightarrow \infty} A^{k} X\). Explain your reasoning clearly.