If spectral radius $\rho(A)<1$ , does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-||A||_{2})$ hold true?

Question

If spectral radius $\rho(A)<1$, does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-\||A||_{2})$ hold true?

If it is correct can somebody give me link to the proof for this inequality?

Answer 1

No, consider $$A=\left[\begin{array}{cc}4/5& 4/5\\0&4/5 \end{array}\right].$$ The spectral radius of $A$ is $4/5$ and its norm is $2/5\sqrt{2(3+\sqrt{5})}>1$ so the quantity on RHS of the proposed inequality is negative, an impossibility.

Answer 2

As others have pointed out, your statement is false ($\rho(A) \le \|A\|$). However, if $\| A\|<1$, one can show that $$ \| (I - A)^{-1}\| \le \frac{1}{1-\|A\|}. $$