Polar decomposition

Representation of invertible matrices as unitaries multiplying a Hermitian operator

This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition.

The core idea behind the construction of the polar decomposition is similar to that used to compute the singular-value decomposition.

Note how, from the above construction, it follows that .

the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined

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

The combination of inversion and Hermite conjugation is chosen so that in the singular value decomposition, the unitary factors remain the same and the iteration reduces to Heron's method on the singular values.