MATRIX THEORY-CHAPTER 5 Proof.For any 1>c>0,there exists a matrix norm.such that p(A)l.p(A)+e.Then p(A)=p(A)≤川Al.≤((A)+e) For the given norm.there exists a constant C such that C-1IB.≤B≤CIB: Hence p(A=p(A)≤川4Ⅲ≤C(P(A)+e)' p(A)≤IA/≤Ck((A)+) Then there exists ko such that when k>ko,1C1+e and 0≤I4/k-p(A)≤(p(A)+2 We finish the proof. ◇ Theorem9.ForA∈Mn,∑e1ak,Ak convergesif∑2，la l for some matrit norm·l Corollary 1.For A=[f,then A is invertible. Proof.We can show that in this case-D-Al<1 where D is the diagonal of A.Hence D-A and A are invertible.MATRIX THEORY - CHAPTER 5 3 Proof. For any 1 >  > 0, there exists a matrix norm 9 · 9?such that ρ(A) ≤ 9A9? ≤ ρ(A) + . Then ρ(A) k = ρ(A k ) ≤ 9A9? ≤ (ρ(A) + ) k For the given norm 9 · 9, there exists a constant C such that C −1 9 B9? ≤ 9B9 ≤ C 9 B9? Hence ρ(A) k = ρ(A k ) ≤ 9A k9 ≤ C(ρ(A) + ) k ρ(A) ≤ 9A k9 1/k ≤ C 1/k(ρ(A) + ) Then there exists k0 such that when k > k0, 1 ≤ C 1/k ≤ 1 +  and 0 ≤ 9A k 9 1/k −ρ(A) ≤ (ρ(A) + 2) We finish the proof. Theorem 9. For A ∈ Mn, P∞ k=1 akAk converges if P∞ k=1 |ak| 9 A9 k for some matrix norm 9 · 9. Example 3. (1) e A = I + P∞ k=1 A k k! converges for all A ∈ Mn (2) (I − A) −1 = I + P∞ k=1 Ak converges if and only if ρ(A) < 1. Corollary 1. For A = [aij ], if |aii| > P j6=i |aij |, then A is invertible. Proof. We can show that in this case 9I − D−1A9∞ < 1 where D is the diagonal of A. Hence D−1A and A are invertible.