Page 208 - Matrices theory and applications

and
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p
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W =
(∈ M p−1 (IR)).
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1 . . 1  10.6. Exercises 191

(c) Show that the eigenvalues of W separate strictly those of W .
n n
3. For a 1 ,... ,a n ∈ IR,with a j =1, form the matrix
j
 
a 1 a 2 a 3 a 4 a n
. .
 . . . 
 a 2 b 2 a 3 . . . . 
.
 
.
 . 
a 3 a 3 b 3  ,
 . . . 
M(a):= 
 . 
 ··· . . 
a 4
 
··· ··· a n
 
a n ··· ··· ··· a n b n
where b j := a 1 + ··· + a j−1 − (j − 2)a j .
(a) Compute the eigenvalues and the eigenvectors of M(a).
(b) We limit ourselves to n-uplets a that belong to the simplex S
defined by 0 ≤ a n ≤ ··· ≤ a 1 and a j =1. Showthat for
j
a ∈ SM(a) is bistochastic and b 2 − a 2 ≤ ··· ≤ b n − a n ≤ 1.
(c) Let µ 1 ,... ,µ n be an n-uplet of elements in [0, 1] with µ n =1.
Show that there exists a unique a in S such that {µ 1 ,... ,µ n }
is equal to the spectrum of M(a) (counting with multiplicity).
(d) Consider the unit sphere Σ of M n(IR), when this space is en-

T
dowed with the norm M 2 = ρ(M M). Show that if P ∈ Σ,
2
then there exists a convex polytope T ,ofdimension (n − 1) ,
included in Σ and containing P. Hint: Use Corollary 5.5.1, with
unitary invariance of the norm · 2 .
4. Show that the cost of an iteration of the QR method for a Hermitian
tridiagonal matrix is 20n + O(1).
5. Show that the reduction to the Hessenberg form (in this case,
3
2
tridiagonal form) of a Hermitian matrix costs 7n /6+ O(n )
operations.
6. (Invariants of the algorithm QR )For M ∈ M n (IR)and 1 ≤ k ≤ n−1,
let us denote by (M) k thematrix of size(n−k)×(n−k) obtained by
deleting the first k rows and the last k columns. For example, (I) 1 is
the Jordan matrix J(0; n − 1). We shall denote also by K ∈ M n (IR)
the matrix defined by k 1n =1 and k ij =0 otherwise.