**Unformatted text preview: **hence
σ (Ak ) = σ (A) for each k. But the process does more than just create a matrix
that is similar to A at each step. The magic lies in the fact that if the process
converges, then limk→∞ Ak = R is an upper-triangular matrix in which the
diagonal entries are the eigenvalues of A. Indeed, if Pk → P, then
Qk = PT−1 Pk → PT P = I
k Y
P IG
R so and Rk = Ak+1 QT → R I = R,
k lim Ak = lim Qk Rk = R, k→∞ k→∞ which is necessarily upper triangular having diagonal entries equal to the eigenvalues of A. However, as is often the case, there is a big gap between theory
and practice, and turning this clever idea into a practical algorithm requires signiﬁcant eﬀort. For example, one obvious hurdle that needs to be overcome is the
fact that the R factor in a QR factorization has positive diagonal entries, so,
unless modiﬁcations are made, the “vanilla” version of the QR iteration can’t
converge for matrices with complex or nonpositive eigenvalues. Laying out all of
the details and analyzing the rigors that constitute the practical implementation
of the QR iteration is tedious and would take us too far astray, but the basic
principals are within our reach. O
C • Hessenberg Matrices. A big step in turning the QR iteration into a practical method is to realize that everything can be done with upper-Hessenberg
matrices. As discussed in Example 5.7.4 (p. 350), Householder reduction
can be used to produce an orthogonal matrix P such that PT AP = H1 ,
and Example 5.7.5 (p. 352) shows that Givens reduction easily produces Copyright c 2000 SIAM Buy online from SIAM
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7.3 Functions of Diagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 537 the QR factors of any Hessenberg matrix. Givens reduction on H1 produces the Q factor of H1 as the transposed product of plane rotations
Q1 = PT PT · · · PTn...

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