Prove that the matrix $(I-A)$ is invertible with $ \| (I-A)^{-1} \| \le \frac{1}{1-a}$.

Question

Suppose that $A$ is an $n \times n$ matrix with $\| A\| \le a <1$. Prove that the matrix $(I-A)$ is invertible with $$ \| (I-A)^{-1} \| \le \frac{1}{1-a}.$$ (The choice of norm does not matter)

Please help me to solve this question, I totally have no idea about

http://math.stackexchange.com/questions/110274/how-can-we-show-that-i-a-is-invertible — Mark Wang, Mar 14 '16 at 18:25

score 1 · Answer 1 · answered Mar 14 '16 at 18:24

1

Hint: Consider $\sum_{n\geq 0} A^n$. show it is the inverse of $(I-A)$ and bound its norm.

answered Mar 14 '16 at 18:24

Tsemo Aristide

87,475

Prove that the matrix $(I-A)$ is invertible with $ \| (I-A)^{-1} \| \le \frac{1}{1-a}$.

1 Answers1