On the ‘Sub-Additivity’ of Principal Eigenvectors

[PoomjaiN]

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.)

 

[PoomjaiN]

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!

:-)