Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS