Polar decomposition

Statement

Let $A$ be an $n \times n$ complex matrix. Then there are a unitary matrix $U$ and a positive semidefinite Hermitian matrix $P$ with $$A = UP.$$ The factor $P$ is unique: $P = (A^* A)^{1/2}$, the square root of $A^* A$. If $A$ is invertible, then $U$ is also unique.

Remarks

It follows from the singular value decomposition $A = W \Sigma V^*$: take $U = W V^*$ and $P = V \Sigma V^*$. For $n = 1$, it is the polar form $z = e^{i\theta} r$ of a complex number. There is also a left polar decomposition $A = P' U$ with $P' = (A A^*)^{1/2}$.

Explanations

L1 human

Every square matrix is a stretch along perpendicular axes, followed by a rotation, possibly with a reflection. This generalises writing a complex number as a positive length times a unit complex number. The stretch part is always unique.