The PCA equation

The PCA method is a bilinear technique which express a data matrix as a matrix product of two matrices:

$${\bf X} = {\bf T}{\bf P}^{T} + {\bf E}$$ (1)

where both the scores (${\bf T}$) and the loadings (${\bf P}$) matrix are orthogonal (the loading matrix vectors are also orthonormal). Thus:

$${\bf t}_i^{T}{\bf t}_j = 0, ~\forall i \neq j$$ (2)

$${\bf p}_i^{T}{\bf p}_j = 0, ~\forall i \neq j$$

in addition ${\bf p}_i^{T}{\bf p}_i = 1$.