Editing Polar decomposition (section)

{{Short description|Representation of invertible matrices as unitary operator multiplying a Hermitian operator}}
In [[mathematics]], the '''polar decomposition''' of a square [[real number|real]] or [[complex number|complex]] [[matrix (mathematics)|matrix]] <math>A</math> is a [[matrix decomposition|factorization]] of the form <math>A = U P</math>, where <math>U</math> is a [[unitary matrix]], and <math>P</math> is a [[positive semi-definite matrix|positive semi-definite]] [[Hermitian matrix]] (<math>U</math> is an [[orthogonal matrix]], and <math>P</math> is a positive semi-definite [[symmetric matrix]] in the real case), both square and of the same size.<ref>{{harvnb|Hall|2015|loc=Section 2.5}}.</ref> 

If a real <math>n \times n</math> matrix <math>A</math> is interpreted as a [[linear transformation]] of <math>n</math>-dimensional [[Cartesian space|space]] <math>\mathbb{R}^n</math>, the polar decomposition separates it into a [[rotation (geometry)|rotation]] or [[reflection (geometry)|reflection]] <math>U</math> of <math>\mathbb{R}^n</math> and a [[scaling (geometry)|scaling]] of the space along a set of <math>n</math> orthogonal axes. 

The polar decomposition of a square matrix <math>A</math> always exists. If <math>A</math> is [[invertible matrix|invertible]], the decomposition is unique, and the factor <math>P</math> will be [[positive-definite matrix|positive-definite]]. In that case, <math>A</math> can be written uniquely in the form <math>A = U e^X</math>, where <math>U</math> is unitary, and <math>X</math> is the unique self-adjoint [[logarithm of a matrix|logarithm]] of the matrix <math>P</math>.<ref>{{harvnb|Hall|2015|loc=Theorem 2.17}}.</ref> This decomposition is useful in computing the [[fundamental group]] of (matrix) [[Lie group]]s.<ref>{{harvnb|Hall|2015|loc=Section 13.3}}.</ref>

The polar decomposition can also be defined as <math>A = P' U</math>, where <math>P' = U P U^{-1}</math> is a symmetric positive-definite matrix with the same eigenvalues as <math>P</math> but different eigenvectors.

The polar decomposition of a matrix can be seen as the matrix analog of the [[complex number#Polar form|polar form]] of a [[complex number]] <math>z</math> as <math>z = u r</math>, where <math>r</math> is its [[absolute value#Complex numbers|absolute value]] (a non-negative [[real number]]), and <math>u</math> is a complex number with unit norm (an element of the [[circle group]]).

The definition <math>A = UP</math> may be extended to rectangular matrices <math>A \in \mathbb{C}^{m \times n}</math> by requiring <math>U \in \mathbb{C}^{m \times n}</math> to be a [[Semi-orthogonal matrix|semi-unitary]] matrix, and <math>P \in \mathbb{C}^{n \times n}</math> to be a positive-semidefinite Hermitian matrix. The decomposition always exists, and <math>P</math> is always unique. The matrix <math>U</math> is unique if and only if <math>A</math> has full rank.<ref name="higham1990"/>