ln by theorem the matrix [ vn of right-eigenvectors is unitary matrix define the -th singular value as si li and suppose lr are all greater than and lr+ ln in particularavi for let ui avi /si thenfor ij rhui * ui * aavi si singular value li { { jsi we can extend ur to an orthonormal basis { um of cm ( using the gramschmidt procedurelet [ un be the corresponding unitary matrix defining to be the diagonal matrix with diagonal ( sr )we haveus [av avr avand hence usvnote that aausvvsuussuand aa vsuusvvssvsou is unitary matrix whose columns are eigenvectors of aaand is unitary matrix whose columns are eigenvectors of aa the svd makes it possible to write the matrix as sum of rank- matricesweighted by the singular values {si } um sr si ui * ( = vn which is called the dyade or spectral representation of for real-valued matricesthe svd has nice geometric interpretationillustrated in figure the linear mapping defined by matrix can be thought of as succession of three linear operations( an orthogonal transformation ( rotation with possible flipping of some axes)corresponding to matrix vfollowed by ( simple scaling of the unit vectorscorresponding to sfollowed by ( another orthogonal transformationcorresponding to - - - - - - - - - - - - figure the figure shows how the unit circle and unit vectors (first panelare first rotated (second panel)then scaled (third panel)and finally rotated and flipped spectral representation