Page 112 - Matrices theory and applications

of the form (i l ,j l )or (i l+1 ,j l ) (compare with the proof of Theorem
5.5.1). Show that in the canonical decomposition of M,the I l are the
equivalence classes of R.
Deduce that the following matrix belongs to S∆ n :
1/21/2
0

.
.


1/2 0 1/2 ··· . 0 . . .  5.6. Exercises 95
 
 
. .
 . .  .

 0 1/2 . . 0
.
 
. . .
 . . 
 . . . 0 1/2 
0 ··· 0 1/21/2

16. Let M ∈ S∆ n and M ∈ ∆ n be given. Show that MM ,M M ∈ S∆ n .
17. If M ∈ S∆ n , show that lim N→+∞ M N exists.
18. Consider the induced norm · p on M n (CC). Let M be a bistochastic
matrix.
(a) Compute M 1 and M ∞.
(b) Show that M ≥ 1 for every induced norm.
(c) Deduce from Theorem 4.3.1 that M p = 1.Towhatextentis
this result different from Corollary 5.5.1?
19. Suppose that we are given three real symmetric matrices (or
Hermitian matrices) A, B, C = A + B.
(a) If t ∈ [0, 1] consider the matrix S(t):= A+tB,so that S(0) = A
and S(1) = C. Arrange the eigenvalues of S(t)inincreasing
order λ 1 (t) ≤ ··· ≤ λ n (t). For each value of t there exists an
orthonormal eigenbasis {X 1 (t),... ,X n (t)}. Weadmit thefact
that it can be chosen continuously with respect to t,so that
t → X j (t) is continuous with a piecewise continuous derivative.

Show that λ (t)=(BX j (t),X j (t)).
j
(b) Let α j ,β j ,γ j (j =1,... ,n) be the eigenvalues of A, B, C,
respectively. Deduce from part (a) that
1
γ j − α j = (BX j (t),X j (t)) dt.
0
(c) Let {Y 1 ,... ,Y n } be an orthonormal eigenbasis, relative to B.
Define
1
2
σ jk := |(X j (t),Y k )| dt.
0
Show that the matrix Σ := (σ jk ) 1≤j,k≤n is bistochastic.
(d) Show that γ j −α j = σ jk β k . Deduce (Lidskii’s theorem) that
k
the vector (γ 1 − α 1 ,... ,γ n − α n ) belongs to the convex hull of
the vectors obtained from the vector (β 1 ,... ,β n ) by all possible
permutations of the coordinates.