Hi Penny,
Is there a reason that you think that a matrix of all ones and zeros can't have a latent value greater than 1? Here is a counterexample:
n = 50;
m = 20;
A = [ones(n,m);triu(ones(m,m));zeros(n,m)];
[coeff,score,latent] = pca(A);
rA =rank(A)
% results
latent =
4.4864
0.2955
0.0826
0.0379
0.0218
0.0143
0.0102
0.0077
0.0061
0.0050
0.0042
0.0036
0.0032
0.0029
0.0026
0.0025
0.0023
0.0022
0.0022
0.0021
rA = 20
The triu matrix was inserted so that every column is linearly independent, which sidesteps a potentially artificial trick situation where a lot of columns are identical. Matrix A has full rank of 20.
pca starts out by taking the mean of each column, so the idea here was to make the excursions from the mean as large as possible. WIth only 1 and 0 avialable, this means creating columns that are half ones and half zeros (or close to it). After that, constructing a bunch of columns that are nearly parallel puts most of the deviation along a single axis.
