Is the spectral radius less than $1$?

Question

Let $A \in \mathbb{C^{n \times n}}$, if $A^k$ tends to $0$ as $k$ tends to infinity, does this imply that the spectral radius is less than $1$?

Answer 1

Let $|| \cdot||$ be any submultiplikative norm on $\mathbb{C^{n \times n}}$ and $r(A)$ the spectral radius of $A \in \mathbb{C^{n \times n}}$. Then we have

$$r(A) \le ||A||.$$

Hence

$$r(A)^n=r(A^n) \le ||A^n||$$

for all $n \in \mathbb N$. With $n \to \infty$ we get $r(A)^n \to 0$, hence $r(A) <1$.