A widely used method for estimating eigenvalues of a general matrix A is the QR algorithm. Under suitable conditions, this algorithm produces a sequence of matrices, all similar to A, that become almost upper triangular, with diagonal entries that approach the eigenvalues of A. The main idea is to factor A (or another matrix similar to A) in the form A = Q, R1, where Q, =Q,' and R, is upper triangular. The factors are interchanged to form A, =R,Q1, which is again factored as A, = Q2R2; then to form A2 = R2Q2, and so on. The similarity of A, A,,.. follows from the more general result below. Show that if A = QR with Q invertible, then A is similar to A, =RQ.
A widely used method for estimating eigenvalues of a general matrix A is the QR algorithm. Under suitable conditions, this algorithm produces a sequence of matrices, all similar to A, that become almost upper triangular, with diagonal entries that approach the eigenvalues of A. The main idea is to factor A (or another matrix similar to A) in the form A = Q, R1, where Q, =Q,' and R, is upper triangular. The factors are interchanged to form A, =R,Q1, which is again factored as A, = Q2R2; then to form A2 = R2Q2, and so on. The similarity of A, A,,.. follows from the more general result below. Show that if A = QR with Q invertible, then A is similar to A, =RQ.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A widely used method for estimating eigenvalues of a general matrix A is the QR algorithm. Under suitable conditions, this algorithm produces a sequence of matrices, all similar to A, that become
almost upper triangular, with diagonal entries that approach the eigenvalues of A. The main idea is to factor A (or another matrix similar to A) in the form A = Q, R,, where Q,
= Q,' and R, is upper
triangular. The factors are interchanged to form A, =R, Q1, which is again factored as A, =Q2R2; then to form A, = R2Q2, and so on. The similarity of A, A, , ... follows from the more general result
below.
Show that if A = QR with Q invertible, then A is similar to A, = RQ.
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