Eigen Values of Positive Definite Matrix

Click For Summary

Discussion Overview

The discussion revolves around methods for finding the eigenvalues of positive definite matrices, particularly in the context of large matrices. Participants explore various algorithms and approaches suitable for different matrix sizes and requirements.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about finding eigenvalues of a positive definite matrix.
  • Another suggests finding the characteristic polynomial and its roots, noting that the roots should be positive.
  • A participant points out that the characteristic polynomial approach applies to any matrix and requests an algorithm suitable for large matrices.
  • Reference is made to the power iteration method as a potential solution for large matrices.
  • Questions arise regarding the definition of "large" matrices, with suggestions of various sizes and whether all eigenvalues or just a few are needed.
  • Discussion includes the EISPACK library for efficient eigenvalue computation for moderate-sized matrices.
  • The Lanczos algorithm is mentioned as a good method for finding some eigenvalues of larger positive definite matrices, with a caution about numerical issues in implementation.
  • QR decomposition is raised as a method, with the assertion that it should work well for a 100x100 matrix.
  • Participants recommend using existing software packages like MATLAB, Maple, and Mathematica, as well as free alternatives like Octave for eigenvalue calculations.

Areas of Agreement / Disagreement

Participants express varying opinions on the best methods for finding eigenvalues, with no consensus on a single approach. Different algorithms and software options are discussed, indicating multiple competing views on the topic.

Contextual Notes

Participants note the importance of matrix size and the specific requirements for eigenvalue computation, which may affect the choice of method. There are also mentions of potential numerical issues with certain algorithms.

vaibhavtewari
Messages
63
Reaction score
0
Hi, i was wondering is there a way I can find the eigenvalues of a positive definite matrix. ?
 
Physics news on Phys.org
Yes, find the characteristic polynomial and find its roots. The roots are the eigenvalues and should be positive.
 
That is true for any matrix, I was hoping to get an algorithm ..my matrix is large
 
How large is "large?" 100x100 or 1 millon x 1 million? Do you want all the eigenvalues, or just a few of them (either the largest or smallest ones?)

There are several different methods that will find all the eigenvalues of a moderate sized matrix (up to about 500 x 500 or 1000 x 1000 if you get lucky) efficiently. Google for the EISPACK library for computer codes. Trying to do this by hand is far too much work for anything bigger than about a 3x3 matrix.

For finding some eigenvalues of bigger positive definite matrices, probably the best method is the Lanczos algorithm, but I woudn't advise trying to write your own code to implement it (even though the maths appears to be quite straightforward) because there are some subtle numerical issues that you have to get right, or else it will fail to converge.
 
What are your thoughts regarding QR decomposition ?
 
my maximum matrix size is 100*100, I want to find all the eigen values.
 
Just use one of the many routines or algorithms available. For example, MATLAB, Maple, and Mathematica are commercial software packages that could do it. Free packages are also available (for example, Octave is a free MATLAB clone). Or if you want an algorithm, use one of the many ones available on the NETLIB site. You could probably find one specific to your problem (for example, eigenvalues only, real entries, symmetric, etc.).
 
vaibhavtewari said:
What are your thoughts regarding QR decomposition ?

Almost any method wiill work on a 100x100 matrix. On a modern PC the run time will be of the order of 1 second or less, so efficiency probably isn't important unless you want to solve thousands of matrices. QR decomposition should work fine.
 

Similar threads

Undergrad Proof about a positive definite matrix
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Why are the eigenvectors of this hermitian matrix not orthogonal?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How can I test for positive semi-definiteness in matrices?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Definiteness of a nonsymmetric matrix
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Eigenvector of Pauli Matrix (z-component of Pauli matrix)
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad In 4x4 matrix when does row swapping not affect eigenvaluess?
  • · Replies 6 ·
Replies
6
Views
4K
Graduate Vector iteration/Power method for eigen values
  • · Replies 6 ·
Replies
6
Views
6K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K