Sorting eigenvectors using symbolic toolbox for PageRank algorithm

I wondering how I can sort the eigenvectors when I am using "d" as a symbolic from the symbolic toolbox? I tried using [U,D]= eig(vpa(M)), but that didn't work. Do I perhaps have to decide how the symbolic roots are sorted when solving for the eigenvectors? I am not quite sure how to do this in Matlab.

The code is given below where I have written a function and what I have in the script separately.

function [Vs] = pagerank_function(linkMatrix,d)
    n = size(linkMatrix,1)
    M = d * linkMatrix + (1-d)/n * ones(n)
    % diagonal matrix eigenvalues D, eigenvectors mtx U
    %[U,D]= eig(vpa(M))
    [U,D] = eig((M))
    [d,ind] = sort(diag(D))
    Ds = D(ind,ind)
    Vs = V(:,ind)
end

Which takes the following matrix

L2 = [0 1/2 1/3 0 0 0 0;
      1/3 0 0 0 1/2 0 0;
      1/3 1/2 0 1 0 0 0;
      1/3 0 1/3 0 1/2 0 0;
      0 0 0 0 0 0 0;
      0 0 1/3 0 0 1 0;
      0 0 0 0 0 0 1];   
   
  
d = sym('d');
            

        
[Vs] = pagerank_function(L2,d);

The general goal I want to achieve is to study how the damping factor 'd' influences all the eigenvectors produced in 'pagerank_function'. Any help would be appreciated as I am quite new to matlab.

Answer 1

[Christine Tobler]

There's an unknown variable in the value you pass to sort, so this won't be sorted by magnitude as the magnitude isn't known.

Here's a small example:

syms a b c d
sort([b c a])

ans = 

So the sort function for arrays of symbolic variables is sorting them lexicographically, not based on value.

That gets us to the key issue with symbolically finding the largest eigenvalue of a matrix: If you have a formula for each of the eigenvalues of a matrix, depending on parameter d, then which of these eigenvalues is largest will depend on that parameter.

For example:

eigvals = eig([d d; 2 -d])

eigvals = 

dvec = -10:10;
plot(dvec, subs(eigvals, dvec))

Warning: Imaginary parts of complex X and/or Y arguments ignored.

So depending on your choice of d, the first or the second of the formulas for the eigenvalue could the the larger one.