Given $I_{n} + A$ is nonsingular, prove:
$||(I_{n} + A)^{-1}||_{p} \le \frac{1}{1-||A||_{p}}$
Now, I know that $I_{n} = (I_{n} + A)^{-1}(I_{n}+A)$, which simplifies to:
$(I_{n} + A)^{-1} = I_{n} - (I_{n}+A)^{-1}A$
Then I can applies the p norms to that,
$||(I_{n} + A)^{-1}||_{p} = ||I_{n} - (I_{n}+A)^{-1}A||_{p}$
$||(I_{n} + A)^{-1}||_{p} \le ||I_{n}||_{p} - ||(I_{n}+A)^{-1}A||_{p}$
$||(I_{n} + A)^{-1}||_{p} \le 1 - ||(I_{n}+A)^{-1}||_{p}||A||_{p}$
but I cannot seem to reduce it down further. What am I missing?
I made an error with my absolute signs, so that is the reason I could not solve it. For those who still may be lost, here is the answer:
$||(I_{n} + A)^{-1}||_{p} = ||I_{n} - (I_{n}+A)^{-1}A||_{p}$
$||(I_{n} + A)^{-1}||_{p} \le ||I_{n}||_{p} **+** ||(I_{n}+A)^{-1}A||_{p}$
$||(I_{n} + A)^{-1}||_{p} \le 1 **+** ||(I_{n}+A)^{-1}||_{p}||A||_{p}$
$||(I_{n} + A)^{-1}||_{p} - ||(I_{n}+A)^{-1}||_{p}||A||_{p}\le 1 $
$||(I_{n} + A)^{-1}||_{p} \le \frac{1}{1-||A||_{p}}$