On the ‘Sub-Additivity’ of Principal Eigenvectors

In summary, the conversation discusses a mathematical problem involving square matrices and their corresponding principal eigenvectors. The question is whether it is possible to find a set of weights that would make the principal eigenvector of the aggregate matrix equal to the weighted sum of the principal eigenvectors of the individual matrices. The question also asks if there is an algorithm to find these weights. The author expresses gratitude for any help in resolving this issue.
  • #1
PoomjaiN
4
0
Dear Friends & Colleagues,

I have a couple of nagging issues with mathematics I was hoping anyone of you would kindly be able to help resolve.

Given a few (let’s say 3) square matrices, A_1, A_2, A_3, and the corresponding principal eigenvector (eigenvector corresponding to the largest eigenvalue, as per eigenvalue decomposition) designated v_1, v_2, v_3. Define the aggregate matrix, A_0, and its corresponding principal eigenvector v_0
I wish to know (i) under which conditions would it be possible to guarantee that there exists a set of weights, w_1, w_2, w_3, such that w_1*v_1 + w_2*v_2 + w_3*v_3 = v_0 (which I am tempted to define as ‘sub-additivity’ vis-à-vis said principal eigenvectors), and (ii) whether an algorithm exists to find these weights, even for a severely restricted case.

Any enlightenment on this issue would be most truly appreciated.

Yours sincerely,
Poomjai Nacaskul (Ph.D.)
 
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  • #2
Forgot to say A_0 = A_1 + A_2 + A_3.

In other words, underwhich condition is the principal eigenvector of the (weighted) sum of matrices equal to the (weighted) sum of principal eigenvectors of the corresponding matrices?

Simple question, yet I haven't been able to find the answer!

:-)
 

1. What is the main concept of "On the ‘Sub-Additivity’ of Principal Eigenvectors"?

The main concept of "On the ‘Sub-Additivity’ of Principal Eigenvectors" is to investigate the behavior of the principal eigenvectors of a matrix when the matrix is decomposed into smaller submatrices.

2. Why is the sub-additivity of principal eigenvectors important?

The sub-additivity of principal eigenvectors is important because it provides insights into the properties and behavior of principal eigenvectors in a decomposed matrix. This can have implications in various fields such as data analysis, signal processing, and machine learning.

3. How is the sub-additivity of principal eigenvectors determined?

The sub-additivity of principal eigenvectors is determined by analyzing the relationship between the principal eigenvectors of a matrix and its submatrices. This can be done through mathematical proofs and numerical experiments.

4. What are the potential applications of the findings in "On the ‘Sub-Additivity’ of Principal Eigenvectors"?

The findings in "On the ‘Sub-Additivity’ of Principal Eigenvectors" can have various applications in fields such as data compression, dimensionality reduction, and feature extraction. It can also aid in understanding the behavior of principal eigenvectors in real-world systems.

5. Are there any limitations to the study of "On the ‘Sub-Additivity’ of Principal Eigenvectors"?

As with any scientific study, there may be limitations to the results and conclusions drawn in "On the ‘Sub-Additivity’ of Principal Eigenvectors". These limitations could include assumptions made in the analysis, sample size, and the specific matrices and submatrices used in the study.

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