TITLE: $\left \| \cdot \right \|$ is an induced norm. If $\left \| A \right \|<1$, how to show that $I-A$ is nonsingular and ...?
QUESTION [5 upvotes]: The induced norm $\left \| \cdot  \right \|$ is defined for a matrix
$A\in\mathbb{C}^{n\times n}$ as $\left \| A \right \|=\sup_{||x||=1} \left \| Ax \right \|$. If $\left \| A \right \|<1$, show that (1) $I-A$ is nonsingular and (2) $\left \| \left ( I-A \right )^{-1} \right \|\leq \frac{1}{1-\left \| A \right \|}$.

REPLY [2 votes]: Referring to a previous problem, for $||A||<1$, we have
$$ (I-A)^{-1} = \sum_{n=0}^{\infty}A^n $$ 
$$\implies  ||(I-A)^{-1}|| = ||\sum_{n=0}^{\infty}A^n||\leq \sum_{n=0}^{\infty}||A^n||\leq \sum_{n=0}^{\infty}||A||^n=\frac{1}{1-||A||}  $$
$$ \implies ||(I-A)^{-1}|| \leq \frac{1}{1-||A||}. $$
Note that, we have used the fact that 
$$ ||A^n||\leq ||A||^n .$$