Need to find eigenvector that corresponds to max eigenvalue

In summary, the author is trying to find the largest real eigenvalue of a 15x15 matrix. He is confused because some of the eigenvalues are complex and so he needs help. The author follows the logic to find the largest real eigenvalue by looking at each column and if there are 2 non-zero values then this is an eigenvalue that is complex, but if there is only 1 non-zero value then this is an eigenvalue that is real.
  • #1
jpildave
5
0
To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue.

Let's say I already have some method that gives me all the eigenvectors and all the eigenvalues. Let's call the eigenvector matrix V (15x15) and the eigenvalues matrix D (15x15 block diagonal).

I'm confused because some of my eigenvalues are complex and so I don't exactly know how to find the maximum one. I believe I have to do something like a^2 + b^2, but this is where I need help. Do I do this per column in D? (I need the exact steps in calculating and then also find the maximum one.)

Thank you.
 
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  • #2
  • #3
HallsofIvy,

Thanks for your response. I wanted to follow-up on something you mentioned... You said that you think I should be looking for the largest real eigenvalue. I didn't know that I should only be limiting the search to the largest _real_ eigenvalue. I think this will simplify things for me, but can you explain why I only need to be concerned with the _real_ eigenvalues? (Sorry, if this is a stupid question.)

Since I already have the eigenvalues and and the eigenvectors, I do not need to code the QR algorithm myself, however, I have an additional question.

In the 15x15 matrix that contains the eigenvalues, does the following logic make sense to you to find the largest real eigenvalue...?

Look at each column in the matrix. If there are 2 non-zero values in the column, then this is an eigenvalue that is complex. Therefore, don't worry about it and go to the next one. But, if you come find a column that only has one non-zero value, this represents an eigenvalue that is real. Therefore, take note of this value and proceed.

Sound good?
 

Related to Need to find eigenvector that corresponds to max eigenvalue

1. What is an eigenvector?

An eigenvector is a vector in a vector space that, when multiplied by a given matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains the same, but its magnitude is multiplied by a constant value.

2. What is the significance of finding an eigenvector that corresponds to the maximum eigenvalue?

Finding an eigenvector that corresponds to the maximum eigenvalue is important because it allows us to identify the direction in which a given matrix has the most significant impact. This can be useful in a variety of applications, such as identifying the principal components in a dataset or determining the dominant mode of vibration in a mechanical system.

3. How do you find the eigenvector that corresponds to the maximum eigenvalue?

To find the eigenvector that corresponds to the maximum eigenvalue, we first need to find the eigenvalues of the matrix. Then, we can use a variety of methods, such as the power method or the inverse iteration method, to find the corresponding eigenvector.

4. Can there be more than one eigenvector that corresponds to the maximum eigenvalue?

Yes, it is possible for a matrix to have multiple eigenvectors that correspond to the maximum eigenvalue. This can occur when the maximum eigenvalue is a repeated root of the characteristic polynomial, meaning it has a multiplicity greater than 1.

5. Why is it important to normalize the eigenvector that corresponds to the maximum eigenvalue?

Normalizing the eigenvector means scaling it to have a unit length, or a magnitude of 1. This is important because it allows us to compare the relative importance of different eigenvectors. Normalization also simplifies the calculations when using the eigenvector for further analysis or applications.

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