score t1(i) = x(i) ⋅ w(1) in the transformed co-ordinates, or as the corresponding vector in the original variables, {x(i) ⋅ w(1)} w(1). Further components The k-th component can be found by subtracting the first k − 1 principal components from X: and then finding the weight vector which extracts the maximum variance from this new data matrix It turns out that this gives the remaining eigenvectors of XTX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of XTX. The k-th principal component of a data vector x(i) can therefore be given as a score tk(i) = x(i) ⋅ w(k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x(i) ⋅ w(k)} w(k), where w(k) is the kth eigenvector of XTX. The full principal components decomposition of X can therefore be given as where W is a p-by-p matrix of weights whose columns are the eigenvectors of XTX. The transpose of W is sometimes called the 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. Covariances XTX itself can be recognized as proportional to the empirical sample covariance matrix of the dataset XT. The sample covariance Q between two of the different principal components over the dataset is given by: where the eigenvalue property of w(k) has been used to move from line 2 to line 3. However eigenvectors w(j) and w(k) corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset. Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix. In matrix form, the empirical covariance matrix for the original variables can be written The empirical covariance matrix between the principal components becomes where Λ is the diagonal matrix of eigenvalues λ(k) of XTX. λ(k) is equal to the sum of the squares over the dataset associated with each component k, that is, λ(k) = Σi tk2(i) = Σi (x(i) ⋅ w(k))2. Dimensionality reduction The