Page 195 - Matrices theory and applications

10. Approximation of Eigenvalues
178
−1
D, which is unitary. We also define
of those of D), and let D 1 be |D|
−1
U.Then D 2 is unitary and the diagonal
D 2 =diag(u jj /|u jj |)and U = D
2
of U is positive real. From the uniqueness of the QR factorization of an

invertible matrix we obtain
k
k
k
P k = QO k D D 2 ,

1
1
which yields 1 k U k =(D D 2 ) −1 T k RD D 2 |D| U ,
= P −1 P k+1 = D −1 D −k O −1 O k+1 D k+1 D 2 ,
Q k
k 2 1 k 1
k
= U k+1 U −1 = D −1 D −k−1 T k+1 RDR −1 T −1 D D 2 .
R k 2 1 1
k k
Since D −k and D k+1 are bounded, we deduce that Q k converges, to D 1 .
1 1

Similarly, R k − R → 0 n ,where
k

R = D −1 D −k RDR −1 D 1 k−1 D 2 . (10.3)
k
1
2
Thefactthatthe matrix R is upper triangular shows that the strict lower

k
triangular part of A k = Q k R k tends to zero (observe that the sequence
(R k ) k∈IN is bounded, because the set of unitary matrices conjugate to A
is bounded). Similarly, the diagonal of R is |D|, which shows that the

k
diagonal of A k converges to D 1 |D| = D.
Remark: Formula (10.3) shows that the sequence A k does not converge,
at least when the eigenvalues have distinct complex arguments. However,
if the eigenvalues have equal complex arguments, for example if they are
real and positive, then D 1 = αI n and R k → T := D −1 R|D|R −1 D 2 ; hence
2
A k converges to αT. Note that this limit is not diagonal in this case.
The situation is especially favorable for tridiagonal Hermitian matrices.
To begin with, we may assume that A is positive definite, up to the change
of A into A + µI n with µ> −ρ(A). Next, we can write A in block-diagonal
form, where the diagonal blocks are tridiagonal irreducible Hermitian ma-
trices. The QR method then treats each block separately. We are thus
reduced to the case of a Hermitian positive definite, tridiagonal and irre-
ducible matrix. Its eigenvalues are real, strictly positive, and simple, from
Proposition 10.1.2: we have λ 1 > ··· >λ n > 0. We can then use the
following statement.
Theorem 10.2.2 Let A ∈ GL n (CC) be an irreducible Hessenberg matrix
whose eigenvalues are of distinct moduli:
|λ 1 | > ··· > |λ n | (> 0).