In reply to Jiangtang Hu.

Thanks for writing. Yes, the terminology is difficult to follow between different authors. I tried to be consistent with SAS documentation and with J. E. Jackson's comprehensive User's Guide to Principal Components, which I consider to be one of the best books on PCA.

In statistics, the word "scores" seems to mean "coefficients that enable you to write one set of variables as a linear combination of a set of basis vectors." Thus "scores" don't have an exact meaning until you specify an exact basis. If you choose V.T for the (orthonormal) basis, then the "scores for A" are U*Sigma, as you say in (3). However, if you choose Sigma.V.T as a basis, then the "scores for A" are just U. Some authors call Sigma*V.T the principal components; others use V.T, which are also called the principal component directions.