Page 93 - Matrices theory and applications

4. Norms
76
(c) If θ ∈ IR, let us denote by U θ the diagonal matrix, whose kth
diagonal term equals exp(ikθ). Show that
2π
1

irθ
U θ AU dθ.
D r (A)=
e
∗
θ
2π
0
(d) Deduce that D r (A) ≤ A .
(e) Let p be an integer between zero and n − 1and r =2p +1. Let
us denote by T r (A) the matrix whose entry of index (j, k)equals
a jk if |k − j|≤ p, and zero otherwise. For example, T 3 (A)is a
tridiagonal matrix. Show that
1 2π
T r (A)= d p (θ)U θ AU dθ,
∗
θ
2π 0
where
p
ikθ
d p (θ)= e
−p
is the Dirichlet kernel.
(f) Deduce that T r (A) ≤ L p A ,where
1 2π
L p = |d p (θ)|dθ
2π 0
is the Lebesgue constant (note: L p =4π −2 log p + O(1)).
(g) Let ∆(A) be the upper triangular matrix whose entries above
the diagonal coincide with those of A.Using thematrix

0 ∆(A) ∗
B = ,
∆(A) 0
show that ∆(A) 2 ≤ L n A 2 (observe that B 2 = ∆(A) 2 ).
(h) What inequality do we obtain for ∆ 0 (A), the strictly upper tri-
angular matrix whose entries lying strictly above the diagonal
coincide with those of A?
n
14. We endow CC with the usual Hermitian structure, so that M n (CC)is
equipped with the norm A = ρ(A A) 1/2 .
∗
Suppose we are given a sequence of matrices (A j ) j∈ZZ in M n (CC)and
1
a summable sequence γ ∈ l (ZZ) of positive real numbers. Assume,
finally, that for every pair (j, k) ∈ ZZ × ZZ,
2
2
A A k ≤ γ(j − k) , A j A ≤ γ(j − k) .
∗
∗
j k
(a) Let F be a finite subset of ZZ.Let B F denote the sum of the
A j ’s as j runs over F.Show that
(B B F ) 2m ≤ card F γ 2m , ∀m ∈ IN.
∗
F 1
(b) Deduce that B F ≤ γ 1.