How to Compute Eigenvalues Using the QR Algorithm?

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SUMMARY

The discussion focuses on computing eigenvalues of a real symmetric matrix using the QR algorithm. The user, Vishal, outlines a planned implementation involving Householder transformations to create a tridiagonal matrix through (n-2) iterations. He details the QR factorization process, where Q is an orthogonal matrix and R is an upper triangular matrix, culminating in the decomposition A = QR. Vishal seeks clarification on how to extract eigenvalues from this process and requests additional references.

PREREQUISITES
  • Understanding of Linear Algebra concepts, specifically eigenvalues and eigenvectors.
  • Familiarity with the QR algorithm and its application in numerical methods.
  • Knowledge of Householder transformations and their role in matrix factorization.
  • Experience with matrix decompositions, particularly in the context of symmetric matrices.
NEXT STEPS
  • Research the process of extracting eigenvalues from the QR factorization.
  • Study the implementation of Householder transformations in numerical computing.
  • Explore the use of libraries such as NumPy for eigenvalue computations in Python.
  • Read about convergence properties of the QR algorithm and its variations.
USEFUL FOR

Students and professionals in mathematics, computer science, and engineering who are interested in numerical methods for linear algebra, particularly those focusing on eigenvalue problems and matrix computations.

svishal03
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I'm attempting to write a code for computing the Eigen values of a real symmetric matrix and I'm using the QR algorithm.I'm referring wiki,Numerical Recipees book and other web serach articles.

This is a part of the self-study course I'm doing in Linear Algebra to upgrde my skills.

My aim is not only getting the algorithm but also understanding Linear algebra and this site is a great help.

As I conclude, following algorithm is being planned by me for implemantation:

1. First and foremost carry out Householder transformation to obtain a tridiagonal matrix from (n-2) householder iterations where n is the size of the square symmetric matrix.

2. During each of the above n-2 iterations, we have Q1,Q2,Q3…..Q(n-2) Householdr matrices


3. We can now obtain Q and R (of QR factorization) where Q is an orthogonal matrix and R is an upper triangular matrix

4. R = Qn-2* Qn-1*……….*Q2*Q1

5. Q = Q1*Q2*………*Qn-2

6. Thus we decompose the original matrix A into A = QR

Am I right above?

I'm not very clear how to get Eigen values following this.Can anyone site a good refernce?

Vishal
 
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Please can anyone help, it is very urgent..
 

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