where formula_72 is diagonal and positive definite, of dimension formula_73, with formula_74 the number of non-zero eigenvalues of formula_69 (which can be shown to verify formula_76). Note that formula_18 is here by definition a matrix whose formula_78-th column is the formula_78-th eigenvector of formula_69, corresponding to the eigenvalue formula_81. Moreover, the formula_82-th column of formula_18, for formula_84, is an eigenvector of formula_69 with eigenvalue formula_86. This can be expressed by writing formula_18 as formula_88, where the columns of formula_89 and formula_90 therefore contain the eigenvectors of formula_69 corresponding to non-zero and zero eigenvalues, respectively. Using this rewriting of formula_18, the equation becomes: