are the vectors obtained from PCA of a non-negative matrix always non-negative?

Question

I'm curious as to whether this is the case, and whether knowing this fact can help us develop better "PCA analogues".

By vectors, I mean the weight vectors obtained from PCA.

Answer 1

Your weight vectors seem to be eigenvectors of a psd matrix, and will hence be orthogonal, and so some of the coefficients of some of them must be negative.

For example, suppose your matrix was $\pmatrix{1&1\\1&1}$, whose transposed eigen vectors are $(1,1)/\sqrt 2$ and $(1,-1)/\sqrt 2$. The main point is that the eigenvectors are orthogonal and non-zero. So their dot product is zero, which typically means there was cancellation of positive and negative numbers in forming the dot product.