Taking the eigenvector corresponding to the largest eigenvalue of a matrix

[Rochelle Zhang]

Dear Statalisters,
I try to compute eigenvector centrality for a 500*500 matrix , the matrix is non-symmetric. By doing google search, I learned that I could use eigensystem () .
Each row and column of my matrix represents an industry, the element of the matrix gives the dollar value of the transaction between a pair of industry. I want to use eigenvector centrality to measure the centralness of the industry node in the network.

A research paper I build my model on states “Eigenvector centrality cannot always be applied to asymmetric adjacency matrices (Bonacich and Lloyd (2001)). For simplicity, we make the adjacency matrix symmetric by taking the maximum value of the upper and lower triangles, as presented in Figure IA.3. Taking the eigenvector corresponding to the largest eigenvalue of this symmetric matrix as the eigenvector centrality, we find centrality measures”.

I am asking here:

1. If eigensystem () is the right function to use for my non-symmetric matrix to get eigenvector centrality, how do I a vector (not a matrix of eigenvector) corresponding to the largest eigenvalue?

An example:

: A = (1, 2 \ 9, 4)
: X = .
: L = .
: eigensystem(A, X, L)
: X
-.316227766 -.554700196 2
-.948683298 .832050294

X is the a matrix of eigenvector, but consider A matrix consists of industry X and Y, I just need to values out of the X matrix to show the importance of each industry.

Thank you,
Rochelle

[Hua Peng (StataCorp)]

For computing the largest eigenvalue and its corresponding eigenvector, you can use mata function eigensystemselecti(), for an example, see

help mf_eigensystemselect##index

[Rochelle Zhang]

Thank you Hua !