Page 191 - Matrices theory and applications

10. Approximation of Eigenvalues
174
A j+1 := R j Q j .Wethenhave
−1
A j+1 = Q
A j Q j ,
j
which shows that A j+1 is unitarily similar to A j . Hence,
−1
A(Q 0 ··· Q j−1 )
A j =(Q 0 ··· Q j−1 )
is conjugate to A by a unitary transformation. (10.2)
Let P j := Q 0 ··· Q j−1 , which is unitary. Since U n is compact, the se-
quence (P j ) j∈IN possesses cluster values. Let P be one of them. Then
A := P −1 AP = P AP is a cluster point of (A j ) j∈IN . Hence, if the se-

∗
quence (A j ) j converges, its limit is unitarily similar to A, hence has the
same spectrum.
This argument shows that in general, the sequence (A j ) j does not con-
verge to a diagonal matrix, because then the eigenvectors of A would be
the columns of P.Inother words, A would have an orthonormal eigenba-
sis. Namely, A would be normal. Except in this special case, one expects
merely that the sequence (A j ) j converges to a triangular matrix, an expec-
tation that is compatible with Theorem 3.1.3. But even this hope is too
optimistic in general. For example, if A is unitary, then A j = A for every j,
with Q j = A and R j = I n ; in that case, the convergence is useless, since the
limit A is not simpler than the data. We shall see later on that the reason
for this bad behavior is that the eigenvalues of a unitary matrix have the
same modulus: The QR method does not do a good job of separating the
eigenvalues of close modulus.
An important case in which a matrix has at least two eigenvalues of
the same modulus is that of matrices with real entries. If A ∈ M n (IR),
then each Q j is real orthogonal, R j is real, and A j is real. This is seen by
induction on j. A limit A will not be triangular if some eigenvalues of A

are nonreal, that is, if A possesses a pair of complex conjugate eigenvalues.
Let us sum up what can be expected in a brave new world. If all the
eigenvalues of A ∈ M n (CC) have distinct moduli, the sequence (A j ) j might
converge to a triangular matrix, or at least its lower triangular part might
converge to
 
λ 1
0
 λ 2 
 . . .  .


 . . . . . . 
0 ··· 0 λ n
When A ∈ M n (IR), one makes the following assumption. Let p be the
number of real eigenvalues and 2q that of nonreal eigenvalues; then there
are p + q distinct eigenvalue moduli. In that case, (A j ) j might converge to
a block-triangular form, the diagonal blocks being 2×2or1×1. The limits
of the diagonal blocks provide trivially the eigenvalues of A.