matrix - Diagonalization of symmetric matrices in julia yields strange results -

i understand going on program in julia. issue following: have symmetric non-negative matrix diagonalize using

egvals, egvecs = eig(h_mat) 

by theorem, matrix should have maximum eigenvalue associated non-negative eigenvector. h_mat has further trick, first column , row has entry filled zeros.

diagonalization yields maximum positive eigenvalue e_max, in fact last eigenvalue because julia arranges eigenvalues in order biggest, eigenvector associated e_max not have entries 0 or positive (i.e. have negative entries)

egvecs[:,end] # or several components ii, egvecs[ii,end]<0 

this matrix don't proper results, instance:

[0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0.0 0.0 2.0 0.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 1.7320508075688774 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 0.0 2.0 0.0 1.414213562373095 0.0 0.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 0.0 0.0 3.0 0.0 2.0 0.0 0.0 0.0 1.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 2.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 3.0 1.7320508075688774 0.0 0.0 0.0 0.0 1.414213562373095 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 3.0 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 1.0 0.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 1.414213562373095 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 1.414213562373095 0.0 0.0 0.0 1.0 0.0 2.0 0.0 0.0 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 1.414213562373095 0.0 0.0 0.0 1.414213562373095 0.0 3.0 1.414213562373095 0.0 1.414213562373095 1.414213562373095 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.7320508075688774 0.0 0.0 0.0 0.0 1.414213562373095 3.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 0.0 1.414213562373095 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 1.414213562373095 0.0 0.0 3.0 1.0 1.7320508075688774 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.414213562373095 2.0 0.0 1.0 3.0 1.7320508075688774 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.7320508075688774 1.7320508075688774 3.0] 

(should 20x20 matrix)

eigenvectors determined proportionality because requirement solve (a - lambda*i)v = 0 , if v1 solves equation, v2 = -v1 does. in symmetric case, standard normalize vectors one, still leaves sign undetermined. therefore theorem using must vector corresponding largest values can chosen such has non-negative elements. indeed, eigenvector corresponding largest eigenvalue of matrix has non-positive elements. julia> eig(a) |> t -> all(t[2][:,indmax(t[1])] .<= 0) true