Prove that $\lim_{k\rightarrow > \infty} \frac{\|A^{k+2}x\|}{\|A^{k}x\|}=\lambda^2$

Assume that $$A \in \mathbb R^{n×n}$$ has $$n$$ linearly independent eigenvectors $$u_1, u_2, . . ., u_n ∈ \mathbb C^n$$ with associated eigenvalues $$λ_1, λ_2, . . ., λ_n$$ with $$λ_1 = λ, λ_2 = −λ$$, for some $$λ \in \mathbb R$$, and $$λ_j \in \mathbb C$$ with $$|λ_j| < λ$$ for $$j = > 3, 4, . . . , n$$. Assume $$x \in \mathbb R^n$$ has a non-zero component in the two-dimensional subspace spanned by the eigenvectors associated with the eigenvalues $$λ$$ and $$−λ$$. Prove that $$\lim_{k\rightarrow > \infty} \frac{\|A^{k+2}x\|}{\|A^{k}x\|}=\lambda^2.$$

My attempt: I suppose this is related to the power iteration method. $$x = c_1u_1+c_2u_2+...+c_nu_n,$$ then $$A^k x = c_1\lambda_1^ku_1+c_2\lambda_2^ku_2+...+c_n\lambda_n^ku_n=\lambda^k(c_1u_1+c_2u_2)+...+c_n\lambda_n^ku_n.$$ How to proceed?

• Just divide the numerator and the denumerator by $\lambda^k$, then you are almost done. – The Pheromone Kid Apr 29 at 13:22