Page 94 - Matrices theory and applications

n
(c) Show (Cotlar’s lemma)thatfor every x, y ∈ CC ,the series

T
y
j∈ZZ
T
is convergent, and that its sum y Ax defines a matrix A ∈
M n (CC)thatsatisfies
A j x 4.6. Exercises 77
A ≤ γ(j).
j∈ZZ
Hint: For a sequence (u j ) j∈ZZ of real numbers, the series
j u j
is absolutely convergent if and only if there exists M< +∞
such that |u j |≤ M for every finite subset F.
j∈F
(d) Deduce that the series A j converges in M n (CC). May one
j
conclude that it converges normally?
15. Let · be an induced norm on M n (IR). We wish to characterize the
matrices B ∈ M n (IR) such that there exist 0 > 0and ω> 0with
(0 < < 0 )=⇒ ( I n − B ≤ 1 − ω ).
(a) For the norm · ∞ ,itis equivalent that B be strictly diagonally
dominant.
(b) What is the characterization for the norm · 1 ?
T
(c) For the norm · 2 , itis equivalentthat B + B be positive
definite.
16. If A ∈ M n (CC)and j =1,... ,n aregiven, wedefine r j (A):=

|a jk |.For i = j,define
k =j
B ij (A)= {z ∈ CC ; |(z − a ii )(z − a jj )|≤ r i (A)r j (A)}.
These sets are Cassini ovals. Finally, let
B(A):= ∪ 1≤i<j≤n B ij (A).
(a) Show that Sp A ⊂B(A).
(b) Show that this result is sharper than Proposition 4.5.1.
(c) When n = 2, show that in fact Sp A is included in the boundary
of B(A).
17. Let B ∈ M n(CC).
(a) Returning to the proof of Theorem 4.2.1, show that for every
n
> 0there exists on CC aHermitian norm · such that for
the induced norm B ≤ ρ(B)+ .
(b) Deduce that ρ(B) < 1 holds if and only if there exists a matrix
∗
A ∈ HPD n such that A − B AB ∈ HPD n .
18. For A ∈ M n (CC), define
:= max |a ij |, δ := min |a ii − a jj |.
i=j i=j