Detected problem when computing eigenvalues using octave eigs function

Question

I need to perform Principal Components Analysis in a dataset, I am using octave 6.4.0, so it should not be a problem related to outdated software. Anyway, the function I am using relies on a very well tested so I proceeded as usual:

1. Compute the covariance matrix.
2. Compute the eigenvalues and eigenvectors, in this case using eigs.
3. Perform the analysis by projecting the centered data (removing mean) using the computed eigenvectors.

And after doing that, I observed weird behavior of the projected data, so I inspected the eigenvectors. What I found was quite shocking. Basically, the returned eigenvectors did not have unitary norm (none of them), which is already a bad symptom. I checked the returned flag, but surprisingly, it was 0. So, no problem detected, although the returned eigenvectors have a problem, obviously. Then, I asked eigs to iteratively compute only the first k eigenvectors, with k = {1, ..., rank(C_{xx})}, where C_{xx} is the covariance matrix. It is the case that this matrix is full-ranked (at least, the returned minimum eigenvalue is way bigger than 0 when k == cols(C_{xx}) == rows(C_{xx}), so it should not be a problem. I have not found any similar problem, and this is the first time I find this problem after applying this technique and using this eigs function numerous times.

Here you can find the code that demonstrates this weird behavior:

for K=1:64
  [V, E] = eigs(templates_train_covariance_matrix, K);
  fprintf(1, 'K = %i; norm(V1) = %f\n', K, norm(V(:,1)));
endfor

K = 1; norm(V1) = 1.000000
K = 2; norm(V1) = 1.000000
K = 3; norm(V1) = 1.000000
K = 4; norm(V1) = 1.000000
K = 5; norm(V1) = 1.000000
K = 6; norm(V1) = 1.000000
K = 7; norm(V1) = 1.000000
K = 8; norm(V1) = 1.000000
K = 9; norm(V1) = 1.000000
K = 10; norm(V1) = 1.000000
K = 11; norm(V1) = 1.000000
K = 12; norm(V1) = 1.000000
K = 13; norm(V1) = 1.000000
K = 14; norm(V1) = 1.000000
K = 15; norm(V1) = 1.000000
K = 16; norm(V1) = 1.000000
K = 17; norm(V1) = 1.000000
K = 18; norm(V1) = 1.000000
K = 19; norm(V1) = 1.000000
K = 20; norm(V1) = 1.000000
K = 21; norm(V1) = 1.000000
K = 22; norm(V1) = 1.000000
K = 23; norm(V1) = 1.000000
K = 24; norm(V1) = 1.000000
K = 25; norm(V1) = 1.000000
K = 26; norm(V1) = 1.000000
K = 27; norm(V1) = 1.000000
K = 28; norm(V1) = 1.000000
K = 29; norm(V1) = 1.000000
K = 30; norm(V1) = 1.000000
K = 31; norm(V1) = 1.000000
K = 32; norm(V1) = 502385811859.998413
K = 33; norm(V1) = 19456763893.455475
K = 34; norm(V1) = 69151.227303
K = 35; norm(V1) = 60981968713.719788
K = 36; norm(V1) = 13353210849.067781
K = 37; norm(V1) = 27614930720.959068
K = 38; norm(V1) = 648885167180.564331
K = 39; norm(V1) = 1078167.777569
K = 40; norm(V1) = 29009353767.082512
K = 41; norm(V1) = 36887852559.354523
K = 42; norm(V1) = 11190161.502876
K = 43; norm(V1) = 67319513050.957047
K = 44; norm(V1) = 26536642028.965473
K = 45; norm(V1) = 21020237947.192379
K = 46; norm(V1) = 79513212336.981018
K = 47; norm(V1) = 31980738852.512493
K = 48; norm(V1) = 4629669543880.510742
K = 49; norm(V1) = 36066934562.318390
K = 50; norm(V1) = 39489763493.994087
K = 51; norm(V1) = 1015181634977.895996
K = 52; norm(V1) = 12591902465.720659
K = 53; norm(V1) = 292456782831.118652
K = 54; norm(V1) = 206117610126.212677
K = 55; norm(V1) = 218166497201.257477
K = 56; norm(V1) = 1059363.341610
K = 57; norm(V1) = 309852.877416
K = 58; norm(V1) = 404729039821.097961
K = 59; norm(V1) = 1289867814341.026123
K = 60; norm(V1) = 35811264888.668854
K = 61; norm(V1) = 72841577175.781281
K = 62; norm(V1) = 19267635.939175
K = 63; norm(V1) = 14730.384760
K = 64; norm(V1) = 87973266862.405182

I can also share the problematic covariance matrix if anyone wants to reproduce this.

Any idea of why this may be happening?