Why does eigenface have at most $N - 1$ non-zero eigenvalues?

Question

I'm reading the wikipedia article on eigenface. It says that if you have $N$ data points, "there will be at most N − 1 eigenvectors with non-zero eigenvalues". Why is this? Why not $N$?

Answer 1

Note that the matrix built from the training samples has had the mean subtracted. That is, if $x_1,\dots,x_N$ are the columns of $T$ and $\mu = \frac{1}{N} (x_1 + \cdots + c_N)$, then the $j$th column of $T$ is $x_j - \mu$.

Verify that the sum of the columns of $T$ is $0$. It follows that $T$ has at least one zero-eigenvalue (corresponding to $T(1,\dots,1)^T = 0$), which means that it has at most $N-1$ non-zero eigenvalues.