Page 207 - Matrices theory and applications

10. Approximation of Eigenvalues
190
When the field is of nonzero characteristic p, the Leverrier method may
p
be employed only if n<p.Since s p = σ , the computation of the s m ’s for
1
m ≥ p does not bring any new information about the σ j ’s.
10.6 Exercises
1. Given a polynomial P ∈ IR[X], use the Euclidean division in order to
define a sequence of nonzero polynomials P j in the following way. Set

P 0 = P, P 1 = P .If P j is not constant, −P j+1 is the remainder of
the division of P j−1 by P j : P j−1 = Q j P j − P j+1 ,deg P j+1 < deg P j .
(a) Assume that P has only simple roots. Show that the sequence
(P j ) j is well-defined, that it has only finitely many terms, and
that it is a Sturm sequence.
(b) Use Proposition 10.1.3 to compute the number of real roots of
2
3
the real polynomials X + aX + b or X + pX + q in terms of
their discriminants.
2. (J. Wilkinson [35], Section 5.45) Let n =2p − 1be an odd number
and W n ∈ M n (IR) be the symmetric tridiagonal matrix
 
p 1
. .
 . . 
 1 . . 
 
. .
 . .  .

 . 1 .
 
 . . . . 
 . . 1 
1 p
The diagonal entries are thus p, p−1,... , 2, 1, 2,... ,p−1,p,and the
subdiagonal entries are equal to 1.
(a) Show that the linear subspace
n
E = {X ∈ IR | x p+j = x p−j , 1 ≤ j< p}
is stable under W n . Similarly, show that the linear subspace
n
E = {X ∈ IR | x p+j = −x p−j , 0 ≤ j< p}
is stable under W n .
(b) Deduce that the spectrum of W n is the union of the spectra of
the matrices
p 1
 
. .
 . . 
1 . .
 
 
W =  . . .  (∈ M p (IR))

n . . .
 . . . 
 
 1 2 1 
2 1