Polar decomposition

Coming from singular value decomposition. There exists a simpler decomposition that the singular value decomposition of any square matrix: polar decomposition. It consists of

where $W$ is an orthogonal matrix (rotation or reflection) and $P$ is a positive definite symmetric matrix. That is, every linear transformation of a vector space can be decomposed as a scale change, not necessarily in the main axis direction, and not necessarily with equal scales, and a rigid transformation.
We can obtain it from the singular value decomposition:

and we take $W=UV$ and $P=V^{\perp }\mathrm{\Sigma }V$.