the columns of show the directions of the principal axes of the ellipsoidand the diagonal elements of indicate the relative magnitudes of the principal axes we see that the first principal component is given by the first column of uand the second principal component by the second column of the projection of the point [ ]onto the -dimensional space spanned by the first principal component [ ]is > [ ]with respect to the basis vector is represented by the number > that isz figure "surfboardellipsoid where one principal axis is significantly larger than the other two pca and singular value decomposition (svdin the setting abovewe did not consider any data set drawn from multivariate pdf the whole analysis rested on linear algebra in principal component analysis (pcawe start with data xn where each is -dimensional pca does not require assumptions how the data were obtainedbut to make the link with the previous sectionwe can think of the data as iid draws from multivariate normal pdf let us collect the data in matrix in the usual waythat isx > xn xn xnd xn the matrix will be the pca' input under this settingthe data consists of points in ddimensional spaceand our goal is to present the data using feature vectors of dimension in accordance with the previous sectionwe assume that underlying distribution of the data has expectation vector in practicethis means that before pca is appliedthe data needs to be centered by subtracting the column mean in every columnxi xi principal component analysis