Let $V$ be a finite dimensional inner product space on $\mathbb{F}$ (where $\mathbb{F}$ can be either $\mathbb{R}$ or $\mathbb{C}$) Let $A$ be a linear operator on $V$.
The polar value decomposition of $A$ is a pair on operators on $V$, $U$ and $P$ with $U$ unitary (i.e. $U^{*}U = I$) and $P$ positive. $P$ is always unique, whereas $P$ is not
I wuould like to gain more insight about this topic. For example, can we say something more in the following cases:
- $A$ normal
- $A$ self-adjoint
- $A$ invertible (this is iff $U$ is unique, but i dont know how to prove it)
You can find answers by reading this excellent wikipedia article: http://en.wikipedia.org/wiki/Polar_decomposition