Linear Algebra, Theory And Applications, 2012a

15.2. THE QR ALGORITHM 395
below the blocks on the main diagonal are small. Then looking at these blocks gives a way
to approximate the eigenvalues. An important example of the concept of a block triangular
matrix is the real Schur form for a matrix discussed in Theorem 7.4.6 but the concept as
described here allows for any size block centered on the diagonal.
First it is important to note a simple fact about unitary diagonal matrices. In what
follows Λ will denote a unitary matrix which is also a diagonal matrix. These matrices
are just the identity matrix with some of the ones replaced with a number of the form e iθ
for some θ. The important property of multiplication of any matrix by Λ on either side
is that it leaves all the zero entries the same and also preserves the absolute values of the
other entries. Thus a block triangular matrix multiplied by Λ on either side is still block
triangular. If the matrix is close to being block triangular this property of being close to a
block triangular matrix is also preserved by multiplying on either side by Λ. Other patterns
depending only on the size of the absolute value occurring in the matrix are also preserved
by multiplying on either side by Λ. In other words, in looking for a pattern in a matrix,
multiplication by Λ is irrelevant.
Now let A be an n × n matrix having real or complex entries. By Lemma 15.2.2 and the
assumption that A is nondefective, there exists an invertible S,
where
A k = Q (k) R (k) = SD k S −1 (15.15)
⎛
⎞
λ 1 0
⎜
D = ⎝
. ..
⎟
⎠
0 λ n
and by rearranging the columns of S, D canbemadesuchthat
Assume S −1 has an LU factorization. Then
|λ 1 |≥|λ 2 |≥···≥|λ n | .
A k = SD k LU = SD k LD −k D k U.
Consider the matrix in the middle, D k LD −k . The ij th entry is of the form
(
D k LD −k) ij = ⎧
⎨
⎩
λ k i L ij λ −k
j
1ifi = j
0ifj>i
and these all converge to 0 whenever |λ i | < |λ j | . Thus
D k LD −k =(L k + E k )
if j