points such that xs- cfor some in particularconsider the ellipsoid xs- principal axes singular value decomposition principal components rd ( let bbwhere is for example the (lowercholesky matrix thenas explained in example the ellipsoid ( can also be viewed as the linear transformation of -dimensional unit sphere via matrix moreoverthe principal axes of the ellipsoid can be found via singular value decomposition (svdof (or )see section and example in particularsuppose that an svd of is udv(note that an svd of is then ud uthe columns of the matrix ud correspond to the principal axes of the ellipsoidand the relative magnitudes of the axes are given by the elements of the diagonal matrix if some of these magnitudes are small compared to the othersa reduction in the dimension of the space may be achieved by projecting each point rd onto the subspace spanned by the main (say dcolumns of -the so-called principal components suppose without loss of generality that the first principal components are given by the first columns of uand let uk be the corresponding matrix with respect to the standard basis {ei }the vector xd ed is represented by the -dimensional vector [ xd ]with respect to the orthonormal basis {ui formed by the columns of matrix uthe representation of is ux similarlythe projection of any point onto the subspace spanned by the first principal vectors is represented by the -dimensional vector > xwith respect to the orthonormal basis formed by the columns of uk sothe idea is that if point lies close to its projection uk > xwe may represent it via numbers instead of dusing the combined features given by the principal components see section for review of projections and orthonormal bases example (principal componentsconsider the matrix which can be written as bbwith figure depicts the ellipsoid xs- which can be obtained by linearly transforming the points on the unit sphere by means of the matrix the principal axes and sizes of the ellipsoid are found through singular value decomposition udvwhere and are - - and -