decomposition. If an unbounded operator A is affiliated to a von Neumann algebra M, and A = UP is its polar decomposition, then U is in M and so is the spectral projection of P, 1B(P), for any Borel set B in . Quaternion polar decomposition The polar decomposition of quaternions H depends on the unit 2-dimensional sphere of square roots of minus one. Given any r on this sphere, and an angle −π < a ≤ π, the versor is on the unit 3-sphere of H. For a = 0 and a = π, the versor is 1 or −1 regardless of which r is selected. The norm t of a quaternion q is the Euclidean distance from the origin to q. When a quaternion is not just a real number, then there is a unique polar decomposition Alternative planar decompositions In the Cartesian plane, alternative planar ring decompositions arise as follows: {{bulleted list | If , is a polar decomposition of a dual number , where ; i.e., ε is nilpotent. In this polar decomposition, the unit circle has been replaced by the line , the polar angle by the slope y/x, and the radius x is negative in the left half-plane. | If , then the unit hyperbola and its conjugate can be used to form a polar decomposition based on the branch of the unit hyperbola through . This branch is parametrized by the hyperbolic angle a and is written where and the arithmetic<ref>Sobczyk, G.(1995) "Hyperbolic Number Plane", College Mathematics Journal 26:268–80</ref> of split-complex numbers is used. The branch through is traced by −eaj. Since the operation of multiplying by j reflects a point across the line , the second hyperbola has branches traced by jeaj or −jeaj. Therefore a point in one of the quadrants has a polar decomposition in one of the forms: The set has products that make it isomorphic to the Klein four-group. Evidently polar decomposition in this case involves an element from that group. }} Numerical determination of the matrix polar decomposition To compute an approximation of the polar decomposition A = UP, usually the unitary factor U is approximated. The iteration is based on Heron's method for the square root of 1'' and computes, starting from , the sequence The combination of inversion and Hermite conjugation is chosen so that in the singular value decomposition, the unitary factors remain the