Polar decomposition

In mathematics, the polar decomposition of a square real or complex matrix ${\displaystyle A}$ is a factorization of the form ${\displaystyle A=UP}$, where ${\displaystyle U}$ is a unitary matrix and ${\displaystyle P}$ is a positive semi-definite Hermitian matrix (${\displaystyle U}$ is an orthogonal matrix and ${\displaystyle P}$ is a positive semi-definite symmetric matrix in the real case), both square and of the same size.^[1]

If a real ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ is interpreted as a linear transformation of ${\displaystyle n}$-dimensional space ${\displaystyle \mathbb {R} ^{n}}$, the polar decomposition separates it into a rotation or reflection ${\displaystyle U}$ of ${\displaystyle \mathbb {R} ^{n}}$, and a scaling of the space along a set of ${\displaystyle n}$ orthogonal axes.

The polar decomposition of a square matrix ${\displaystyle A}$ always exists. If ${\displaystyle A}$ is invertible, the decomposition is unique, and the factor ${\displaystyle P}$ will be positive-definite. In that case, ${\displaystyle A}$ can be written uniquely in the form ${\displaystyle A=Ue^{X}}$, where ${\displaystyle U}$ is unitary and ${\displaystyle X}$ is the unique self-adjoint logarithm of the matrix ${\displaystyle P}$.^[2] This decomposition is useful in computing the fundamental group of (matrix) Lie groups.^[3]

The polar decomposition can also be defined as ${\displaystyle A=P'U}$ where ${\displaystyle P'=UPU^{-1}}$ is a symmetric positive-definite matrix with the same eigenvalues as ${\displaystyle P}$ but different eigenvectors.

The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number ${\displaystyle z}$ as ${\displaystyle z=ur}$, where ${\displaystyle r}$ is its absolute value (a non-negative real number), and ${\displaystyle u}$ is a complex number with unit norm (an element of the circle group).

The definition ${\displaystyle A=UP}$ may be extended to rectangular matrices ${\displaystyle A\in \mathbb {C} ^{m\times n}}$ by requiring ${\displaystyle U\in \mathbb {C} ^{m\times n}}$ to be a semi-unitary matrix and ${\displaystyle P\in \mathbb {C} ^{n\times n}}$ to be a positive-semidefinite Hermitian matrix. The decomposition always exists and ${\displaystyle P}$ is always unique. The matrix ${\displaystyle U}$ is unique if and only if ${\displaystyle A}$ has full rank. ^[4]