Page 210 - Matrices theory and applications

8. In the Jacobi method, show that if the eigenvalues are simple, then
1
m
the product R ··· R
converges, to an orthogonal matrix R such
∗
that R AR is diagonal.
9. Extend the Jacobi method to Hermitian matrices. Hint: Replace the
rotation matrices

cos θ sin θ 10.6. Exercises 193
− sin θ cos θ
by unitary matrices

z 1 z 2
.
z 3 z 4
10. Let A ∈ Sym (IR) be a matrix whose eigenvalues, of course real, are
n
simple. Apply the Jacobi method, but selecting the angle θ k so that
π/4 ≤|θ k |≤ π/2.
(a) Show that E k tends to zero, that the sequence D k is relatively
compact, and that its cluster values are diagonal matrices whose
diagonal terms are the eigenvalues of A.
(b) Show that an iteration has the effect of permuting, asymp-
(k) (k)
totically, a pp and a qq ,where (p, q)=(p k ,q k ). In other
words
lim |a (k+1) − a (k) | =0,
pp
qq
k→+∞
and vice versa, permuting p and q.
11. The Bernoulli method computes an approximation of the root of
n
largest modulus for a polynomial a 0 X + ··· + a n , when that root
is unique. To do so, one defines a sequence by a linear induction of
order n:
1
z k = − (a 1 z k−1 + ··· + a n z k−n ).
a 0
Compare this method with the power method for a suitable matrix.
12. Consider the power method for a matrix M ∈ M n (CC)ofwhich several
eigenvalues are of modulus ρ(M) = 0. Again, CC n = E ⊕ F is the
n
decomposition of CC into linear subspaces stable under M,suchthat
0
ρ(M| F ) <ρ(M)and λ ∈ Sp(M| E )=⇒|λ| = ρ(M). Finally, x =
0
0
0
0
0
y + z with y ∈ E, z ∈ F,and y =0.
(a) Express
m−1
1 k
log Mx
m
k=0
m 0
in terms of M x .