Indeed, they each arise in polar decomposition of a complex number z.

The last condition follows by uniqueness of the polar decomposition.

An appeal to polar decomposition extend this to the general case where T need not be positive.

Evidently polar decomposition in this case involves an element from that group.

Notice that the polar decomposition of an invertible matrix is unique.

To conserve momentum the rotation of the body must be estimated properly, for example via polar decomposition.

This parameter is part of the polar decomposition of a split-complex number.

Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition.

This polar decomposition is unique as is non-symmetric.

It generalizes the polar decomposition of matrices.

The existence of a polar decomposition is a consequence of Douglas' lemma:

Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition.

Applying the lemma gives polar decomposition.

See polar decomposition of a quaternion for details of this development of the three-sphere.

One can use the Radon-Nikodym theorem to prove that the variation is a measure and the existence of the polar decomposition.

Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix.

Left polar decomposition is also known as reverse polar decomposition.

When M is non-singular, the Q and S factors of the polar decomposition are uniquely determined.

The polar decomposition of quaternions H depends on the sphere of square roots of minus one.

Therefore a point in one of the quadrants has a polar decomposition in one of the forms:

The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.

From the polar decomposition theorem, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation.

Via polar decomposition, one can prove that the space of pth Schatten class operators is an ideal in B(H).

Similar to the ordinary complex plane, a point, not on the diagonals, has a polar decomposition using the parametrization of the unit hyperbola and the alternative radial length.

Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues.