44
Lemma 5.2.2 Let A1 be the matrix described in Lemma 5.2.1. Let T be the n x n
tridiagonal matrix with O's on the diagonal, with 1's on the superdiagonal, and with
-1's on the subdiagonal. Then IIAI-'TIoo = n 1.
Proof: Since
(AI-IT)i,j = E(Aj-1)i,(W)kj
k=1
= (a-1)i,j-tj-l,j + (a-1)ij+ltj+l,j
= (a-)i-l (a -)i,j+1
we have
(Ai-T)i,j <
Then, the infinity norm for
IIAr1'TIK
-2(n+l-i) ifj=l,2,.i -1
-n + 2i 1i)
if 3 = i
-2, if j =i+1,i + 2,-- ,n-1
n+1
2i
2i if j = n
n+1'
the matrix Ai-'T is
n
= maxE (A-1'T)i,3
j=l
[2(n- i + ~1) -n+2i-1 2i ]
= max (i 1)+ + (n- i)
i n+1 n+1 n+1 J
S -4i2 + 4i(n + 1)- 2(n+ 1) 12i- (n + 1) (5.7)
Smax + -- (5.7
Now focus on the term inside brackets of (5.7):