Bounding the norm of matrix powers

Question

Given two square matrices $A$ and $C \in \mathbb{R}^{n \times n}$ and $\Vert A - C \Vert_2 \leq \beta$ $ \forall \beta \geq 0$, can we say anything about the upper bound of $\Vert A^ {k} - C^{k} \Vert_2$ interms of $\beta$ where $k \in \mathbb{Z^+}$. Also what is the condition for $\rho(C) < 1$, given that $\rho(A) < 1$, where $\rho$ represents the absolute maximum eigen value (spetral radius)?

Answer 1

You can write $$ A^k- C^k = (A-C)A^{k-1} + C(A-C)A^{k-2} + \dots = \sum_{j=0}^k C^j(A-C)A^{k-j-1}. $$ Hence $$ \|A^k-C^k\|_2 \le \|A-C\|_2 \sum_{j=0}^k \|C^j\|_2\|A^{k-j-1}\|_2. $$