Solving an Eigenvalue Problem for Large n Matrix

Click For Summary
SUMMARY

The discussion focuses on solving an eigenvalue problem for a real symmetric matrix A, defined by the equation Ax = λx, where A is an n*n matrix and n is large. Participants confirm that the eigenvalues of A are equal to those of A^T, as expressed by the determinant condition det(A - λI) = det(A^T - λI). While the eigenvectors of symmetric matrices are not guaranteed to be the same as those of non-symmetric matrices, the discussion suggests that Laplace expansions may yield equal eigenvectors for symmetric matrices. Recommended resources include Gilbert Strang's materials from MIT for further study.

PREREQUISITES
  • Understanding of eigenvalue problems and the characteristic equation
  • Familiarity with symmetric matrices and their properties
  • Knowledge of determinants and matrix operations
  • Basic concepts of linear algebra, particularly eigenvalues and eigenvectors
NEXT STEPS
  • Study Gilbert Strang's linear algebra materials, particularly on eigenvalues and eigenvectors
  • Learn about the implications of the spectral theorem for symmetric matrices
  • Explore Laplace expansions and their applications in finding eigenvalues
  • Investigate numerical methods for computing eigenvalues of large matrices
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are dealing with linear algebra concepts, particularly those focused on eigenvalue problems and symmetric matrices.

Pyrokenesis
Messages
19
Reaction score
0
I am having trouble with the following question. (Just hoping to get some guidance, recommended texts etc.):

"Consider an eigenvalue problem Ax = λx, where A is a real symmetric n*n matrix, the transpose of the matrix coincides with the matrix, (A)^T = A. Find all the eigenvalues and all the eigenvectors. Assume that n is a large number."

Any help would be fantastic!
 
Physics news on Phys.org
Originally posted by Pyrokenesis

"Consider an eigenvalue problem Ax = λx, where A is a real symmetric n*n matrix, the transpose of the matrix coincides with the matrix, (A)^T = A. Find all the eigenvalues and all the eigenvectors. Assume that n is a large number."

Don't know how much help I can be, but since I am studying the same material at the moment, I will help with what I can.

The eigenvalues of A are equal to the eigenvalues of A^T because det(A-λI)=det(A^T-λI). The diagonal/trace stays the same here.

(I am guessing on the next part, as our book does not cover this)
Normally, for non-symmetric matrices the eigenvectors are not the same. However, in your case, since A^T=A then (and I am guessing here) I would tend to believe that Laplace expansions would end up yielding equal eigenvectors as well.
 


Originally posted by samoth
Don't know how much help I can be, but since I am studying the same material at the moment, I will help with what I can.

The eigenvalues of A are equal to the eigenvalues of A^T because det(A-λI)=det(A^T-λI). The diagonal/trace stays the same here.

(I am guessing on the next part, as our book does not cover this)
Normally, for non-symmetric matrices the eigenvectors are not the same. However, in your case, since A^T=A then (and I am guessing here) I would tend to believe that Laplace expansions would end up yielding equal eigenvectors as well.

If A^T = A then A is symmetric, no? Does this help?
 
Yes, A is symmetric when A^T=A.

First of all, I was wrong about the eigenvectors of A and A^T being the same. They are not. However, I cannot help as to why, as our text offers only two sentences in this matter. Further, I do not believe I am at a level of knowledge upon which speculation would prove fruitful. Hmm.. let's see. I can give you some links that will hopefully be of some help.

We are using a book by Gilbert Strang from MIT. He has quite a bit of information on his/the books website, as well as fully recorded lectures. Here is his site.


http://web.mit.edu/18.06/www/

I am sorry I can be of little help with this, but I hope this can help you shed some light on the problem.
 
Last edited by a moderator:
Thanx bro,

I know, its a tough subject, cheers for the link. Now that all other coursework is out of the way I will crack on with this and post my findings when I find something.

Good luck with your course as well,

cheers,

Dexter
 
Originally posted by Pyrokenesis

I will crack on with this and post my findings when I find something.

Please do, as I am quite curious now.
Good luck as well with your course!
 

Similar threads

Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Eigenvalues of block matrix/Related non-linear eigenvalue problem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proof about a positive definite matrix
  • · Replies 12 ·
Replies
12
Views
2K
Graduate Quadratic eigenvalue problem and solution (solved in Mathematica)
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Numerically Calculating Eigenvalues
  • · Replies 12 ·
Replies
12
Views
2K
Graduate Creating a matrix with desirable eigenvalues
  • · Replies 7 ·
Replies
7
Views
3K
MHB Help! Wrong Eigenvalues Using Matrix A
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Is a symmetric matrix with positive eigenvalues always real?
  • · Replies 8 ·
Replies
8
Views
3K
MHB Prove matrix has all real eigenvalues
  • · Replies 12 ·
Replies
12
Views
4K