The PCA equation

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

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

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


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

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



Bjørn Alsberg 2004-10-22