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Use Conjugate Gradient to obtain the Eigenvalues

1

$begingroup$

So I have been told that we can use CG to obtain an eigenvalue approximation to the true matrix. I am not sure how? (Connection to Lanczos)

Furthermore I have been told that there is a deep connection between CG and Lanczos. I would really appreciate if someone can clarify how are they connected. I do understand that if we do Lanczos we would get a good approximation to the eigenvalues, but I do not see how can we use CG and get the eigenvalues. Thank you guys!

linear-algebra matrices optimization eigenvalues-eigenvectors numerical-optimization

share|cite|improve this question

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you assume symmetry of the matrix?
  $endgroup$
  – amsmath
  Mar 28 at 2:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I assume matrix meets all the criteria for CG, which includes symmetry.
  $endgroup$
  – Kori
  Mar 28 at 4:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, Lanczos works for rectangular matrices as well.
  $endgroup$
  – amsmath
  Mar 28 at 4:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Good point, but I am looking at symmetric.
  $endgroup$
  – Kori
  Mar 28 at 4:42
  

  
  
  
  

add a comment | 

1

$begingroup$

So I have been told that we can use CG to obtain an eigenvalue approximation to the true matrix. I am not sure how? (Connection to Lanczos)

Furthermore I have been told that there is a deep connection between CG and Lanczos. I would really appreciate if someone can clarify how are they connected. I do understand that if we do Lanczos we would get a good approximation to the eigenvalues, but I do not see how can we use CG and get the eigenvalues. Thank you guys!

linear-algebra matrices optimization eigenvalues-eigenvectors numerical-optimization

share|cite|improve this question

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you assume symmetry of the matrix?
  $endgroup$
  – amsmath
  Mar 28 at 2:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I assume matrix meets all the criteria for CG, which includes symmetry.
  $endgroup$
  – Kori
  Mar 28 at 4:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, Lanczos works for rectangular matrices as well.
  $endgroup$
  – amsmath
  Mar 28 at 4:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Good point, but I am looking at symmetric.
  $endgroup$
  – Kori
  Mar 28 at 4:42
  

  
  
  
  

add a comment | 

$begingroup$

So I have been told that we can use CG to obtain an eigenvalue approximation to the true matrix. I am not sure how? (Connection to Lanczos)

Furthermore I have been told that there is a deep connection between CG and Lanczos. I would really appreciate if someone can clarify how are they connected. I do understand that if we do Lanczos we would get a good approximation to the eigenvalues, but I do not see how can we use CG and get the eigenvalues. Thank you guys!

linear-algebra matrices optimization eigenvalues-eigenvectors numerical-optimization

share|cite|improve this question

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

$endgroup$

share|cite|improve this question

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

share|cite|improve this question

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

edited Mar 28 at 6:30

Rodrigo de Azevedo

13k41960

asked Mar 28 at 2:03

KoriKori

6631519

asked Mar 28 at 2:03

KoriKori

6631519