I am given a lemma and it states:
Lemma: Let A be a real n x n matrix. Then given any e > 0, there is a norm such that
norm(A) <= p(A) + e Where p(A) is the spectral radius of A.
They then go on to state that based on this lemma,
if p(A) < 1, then norm(A) < 1 for the correct choice of the norm.
What I'm looking for help on is understanding how they came to that conclusion based only on that lemma. For example, given
e = .001, then the lemma states there is a norm such that
norm(A) <= p(A) + .001
Since it is a "or equal" then this means there is the possibility that
norm(A) = p(A) + .001, and in such a case then
norm(A) > p(A). So it seems that based on the lemma, then
p(A) < 1 does not necessarily always imply that
norm(A) < 1.
I even tried to do a proof of this but still came up with the same result:
p(A) < 1 -> p(A) + e < 1 + e -> norm(A) <= p(A) +e < 1 + e norm(A) - e <= p(A) < 1
This pretty much says the same thing, if norm(A) - e <= p(A), then p(A) < 1, but if p(A) close to 1 by a difference less than e, then norm(A) > 1 in the "equal to" case where norm(A) - e = p(A)
Now of course you could pick a smaller e, but the way the lemma reads it says that for a given e, you can find a norm s.t. A <= p(A) + e, so the "or equal" part will still get you using a counter example I gave similar to the e = .001 above
Now I have seen a different proof on wikipedia that shows if p(A) < 1 then norm(A) < 1, so I am not disputing that fact, but that proof gets into looking at entries of the Jordan Normal Form. So my point is, yes p(A) < 1 then norm(A) < 1, BUT I don't understand how you can conclude that based only on the Lemma I gave at the beginning. It's these big jumps in rational that confuse me.