Eigen Values of Positive Definite Matrix

In summary: Additionally, methods like Jacobi and Householder transformations can also be used. In summary, there are several methods available such as EISPACK library, Lanczos algorithm, and QR decomposition that can efficiently find all the eigenvalues of a moderate sized matrix. For larger matrices, using software packages or algorithms from sites like NETLIB would be more suitable.
  • #1
vaibhavtewari
65
0
Hi, i was wondering is there a way I can find the eigenvalues of a positive definite matrix. ?
 
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  • #2
Yes, find the characteristic polynomial and find its roots. The roots are the eigenvalues and should be positive.
 
  • #3
That is true for any matrix, I was hoping to get an algorithm ..my matrix is large
 
  • #5
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 postive 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.
 
  • #6
What are your thoughts regarding QR decomposition ?
 
  • #7
my maximum matrix size is 100*100, I want to find all the eigen values.
 
  • #8
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.).
 
  • #9
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.
 

What is a positive definite matrix?

A positive definite matrix is a square matrix where all of its eigenvalues are positive. In other words, when the matrix is multiplied by any non-zero vector, the result is always a positive value.

What are eigenvalues and eigenvectors?

Eigenvalues are scalar values that represent how a linear transformation affects a vector. Eigenvectors are non-zero vectors that, when multiplied by a matrix, result in a scalar multiple of itself, i.e. the vector's direction remains unchanged.

How do you find the eigenvalues of a positive definite matrix?

The eigenvalues of a positive definite matrix can be found by solving the characteristic equation, which is det(A - λI) = 0, where A is the matrix and λ is the eigenvalue. The solutions to this equation are the eigenvalues of the matrix.

What is the significance of positive eigenvalues in a positive definite matrix?

Positive eigenvalues in a positive definite matrix indicate that the matrix is a local minimum, meaning that the matrix has a smaller value at its center than at any other nearby point. This is useful in optimization problems, where we want to find the minimum value of a function.

How are positive definite matrices used in real-world applications?

Positive definite matrices have many applications in various fields such as physics, engineering, and computer science. They are used in optimization problems, machine learning algorithms, and in the analysis of physical systems such as quantum mechanics. They are also used in solving systems of linear equations and in diagonalizing matrices.

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