On the ‘Sub-Additivity’ of Principal Eigenvectors

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SUMMARY

The discussion centers on the conditions necessary for the principal eigenvector of an aggregate matrix A_0, defined as the sum of square matrices A_1, A_2, and A_3, to equal a weighted sum of the principal eigenvectors v_1, v_2, and v_3 of the individual matrices. The author, Poomjai Nacaskul, seeks clarity on the existence of weights w_1, w_2, and w_3 that satisfy the equation w_1*v_1 + w_2*v_2 + w_3*v_3 = v_0. Additionally, the discussion explores whether an algorithm exists to determine these weights under restricted conditions.

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  • Familiarity with linear algebra concepts, particularly matrix addition
  • Knowledge of weighted sums and their implications in vector spaces
  • Basic understanding of algorithms related to matrix computations
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Mathematicians, researchers in linear algebra, and anyone involved in eigenvalue problems or matrix theory will benefit from this discussion.

PoomjaiN
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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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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!

:-)
 

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