To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere of radius one in . The linear map maps this sphere onto an ellipsoid in . Non-zero singular values are simply the lengths of the semi-axes of this ellipsoid. Especially when , and all the singular values are distinct and non-zero, the SVD of the linear map can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid and specifically its axes; then consider the directions in sent by onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry sending these directions to the coordinate axes of . On a second move, apply an endomorphism diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of as stretching coefficients. The composition then sends the unit-sphere onto an ellipsoid isometric to . To define the third and last move , apply an isometry to this ellipsoid so as to carry it over . As can be easily checked, the composition coincides with .