These electronic switches are allowed to change state

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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 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy from 7.3 Functions of Diagonalizable Matrices 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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