I am computing the principle component matrix of a financial database and to obtain the second factor I extrapolate the second vector. So far, it's easy, but I wonder, do the signs of the second component have to be different? I mean can there be only positive (or negative) values ? Regards
Every subsequent eigenvector from a PCA is constrained to be orthogonal to all previous eigenvectors. If your variables tend to be mainly positively correlated, this will lead to an all-positive first eigenvector. In order to be orthogonal to that the second eigenvector will have to have a mixture of signs.