Linear Algebra, Theory And Applications, 2012a

15.2. THE QR ALGORITHM 391
where
⎛
⎞
λ 1 0
⎜
D = ⎝
. ..
⎟
⎠
0 λ n
and suppose S −1 has an LU factorization. Then the matrices A k in the QR algorithm
described above converge to an upper triangular matrix T ′ having the eigenvalues of A,
λ 1 , ··· ,λ n descending on the main diagonal. The matrices Q (k) converge to Q ′ ,anorthogonal
matrix which equals Q except for possibly having some columns multiplied by −1 for Q
the unitary part of the QR factorization of S,
S = QR,
and
lim A k = T ′ = Q ′T AQ ′
k→∞
Proof: From Lemma 15.2.2
A k = Q (k) R (k) = SD k S −1 (15.12)
Let S = QR where this is just a QR factorization which is known to exist and let S −1 = LU
which is assumed to exist. Thus
Q (k) R (k) = QRD k LU (15.13)
and so
Q (k) R (k) = QRD k LU = QRD k LD −k D k U
That matrix in the middle, D k LD −k satisfies
(
D k LD −k) = ij λk i L ij λ −k
j for j ≤ i, 0ifj>i.
Thus for jλ i when this happens. When
i = j it reduces to 1. Thus the matrix in the middle is of the form
where E k → 0. Then it follows
I + E k
A k = Q (k) R (k) = QR (I + E k ) D k U
= Q ( I + RE k R −1) RD k U ≡ Q (I + F k ) RD k U
where F k → 0. Then let I + F k = Q k R k wherethisisanotherQR factorization. Then it
reduces to
Q (k) R (k) = QQ k R k RD k U
This looks really interesting because by Lemma 15.2.3 Q k → I and R k → I because
Q k R k =(I + F k ) → I. So it follows QQ k is an orthogonal matrix converging to Q while
R k RD k U
(R (k)) −1
is upper triangular, being the product of upper triangular matrices. Unfortunately, it is not
known that the diagonal entries of this matrix are nonnegative because of the U. LetΛbe
just like the identity matrix but having some of the ones replaced with −1 insuchaway