Editing Principal component analysis (section)

=== Further components ===
The ''k''-th component can be found by subtracting the first ''k''&nbsp;−&nbsp;1 principal components from '''X''':

:<math>\mathbf{\hat{X}}_k = \mathbf{X} - \sum_{s = 1}^{k - 1} \mathbf{X} \mathbf{w}_{(s)} \mathbf{w}_{(s)}^{\mathsf{T}} </math>

and then finding the weight vector which extracts the maximum variance from this new data matrix
:<math>\mathbf{w}_{(k)}
= \mathop{\operatorname{arg\,max}}_{\left\| \mathbf{w} \right\| = 1} \left\{ \left\| \mathbf{\hat{X}}_{k} \mathbf{w} \right\|^2 \right\}
= \arg\max \left\{ \tfrac{\mathbf{w}^\mathsf{T} \mathbf{\hat{X}}_{k}^\mathsf{T} \mathbf{\hat{X}}_{k} \mathbf{w}}{\mathbf{w}^T \mathbf{w}} \right\}</math>

It turns out that this gives the remaining eigenvectors of '''X'''<sup>T</sup>'''X''', with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of '''X'''<sup>T</sup>'''X'''.

The ''k''-th principal component of a data vector '''x'''<sub>(''i'')</sub> can therefore be given as a score ''t''<sub>''k''(''i'')</sub> = '''x'''<sub>(''i'')</sub> ⋅ '''w'''<sub>(''k'')</sub> in the transformed coordinates, or as the corresponding vector in the space of the original variables, {'''x'''<sub>(''i'')</sub> ⋅ '''w'''<sub>(''k'')</sub>} '''w'''<sub>(''k'')</sub>, where '''w'''<sub>(''k'')</sub> is the ''k''th eigenvector of '''X'''<sup>T</sup>'''X'''.

The full principal components decomposition of '''X''' can therefore be given as
:<math>\mathbf{T} = \mathbf{X} \mathbf{W}</math>
where '''W''' is a ''p''-by-''p'' matrix of weights whose columns are the eigenvectors of '''X'''<sup>T</sup>'''X'''. The transpose of '''W''' is sometimes called the [[whitening transformation|whitening or sphering transformation]]. Columns of '''W''' multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called ''loadings'' in PCA or in Factor analysis.