Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|\_{2} \leq 1/(1-\|A\|\_{2})$ hold true?


$\|A\|\_{2}$ denotes maximum eigenvalue of A.


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



Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?
Since $A$ is positive semidefinite, norm of $A$ is the same as spectral radius of $A$. Thus


$$\|(I-A)^{-1}\|=\|I+A+A^2+\dots\|\leqslant 1+\|A\|+\|A\|^2+\dots=\frac1{1-\|A\|}.$$


