Polar decomposition

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the polar decomposition of a square <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Matrix_decomposition/xVBnAg0x">factorization</a> of the form 
 
 
 
 A
 =
 U
 P
 
 
 {\displaystyle A=UP}
 
, where 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a>, and 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is a <a href="/facts/Positive_semi-definite_matrix/Hbr6AuPS">positive semi-definite</a> <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> (
 
 
 
 U
 
 
 {\displaystyle U}
 
 is an <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a>, and 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is a positive semi-definite <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> in the real case), both square and of the same size. 
If a real 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is interpreted as a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> of 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional <a href="/facts/Cartesian_space/lkTqvMmB">space</a> 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
, the polar decomposition separates it into a <a href="/facts/Rotation_(geometry)/6gtkkcy0">rotation</a> or <a href="/facts/Reflection_(geometry)/5JHWnrql">reflection</a> 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 and a <a href="/facts/Scaling_(geometry)/pFm6fvpu">scaling</a> of the space along a set of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 orthogonal axes. 
The polar decomposition of a square matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 always exists. If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a>, the decomposition is unique, and the factor 
 
 
 
 P
 
 
 {\displaystyle P}
 
 will be <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive-definite</a>. In that case, 
 
 
 
 A
 
 
 {\displaystyle A}
 
 can be written uniquely in the form 
 
 
 
 A
 =
 U
 
 e
 
 X
 
 
 
 
 {\displaystyle A=Ue^{X}}
 
, where 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is unitary, and 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the unique self-adjoint <a href="/facts/Logarithm_of_a_matrix/lhmcCLJh">logarithm</a> of the matrix 
 
 
 
 P
 
 
 {\displaystyle P}
 
. This decomposition is useful in computing the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of (matrix) <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>.
The polar decomposition can also be defined as 
 
 
 
 A
 =
 
 P
 ′
 
 U
 
 
 {\displaystyle A=P'U}
 
, where 
 
 
 
 
 P
 ′
 
 =
 U
 P
 
 U
 
 −
 1
 
 
 
 
 {\displaystyle P'=UPU^{-1}}
 
 is a symmetric positive-definite matrix with the same eigenvalues as 
 
 
 
 P
 
 
 {\displaystyle P}
 
 but different eigenvectors.
The polar decomposition of a matrix can be seen as the matrix analog of the <a href="/facts/Complex_number/2w7mqI7e">polar form</a> of a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> 
 
 
 
 z
 
 
 {\displaystyle z}
 
 as 
 
 
 
 z
 =
 u
 r
 
 
 {\displaystyle z=ur}
 
, where 
 
 
 
 r
 
 
 {\displaystyle r}
 
 is its <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> (a non-negative <a href="/facts/Real_number/R02gw5Pb">real number</a>), and 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is a complex number with unit norm (an element of the <a href="/facts/Circle_group/kuzdWtHo">circle group</a>).
The definition 
 
 
 
 A
 =
 U
 P
 
 
 {\displaystyle A=UP}
 
 may be extended to rectangular matrices 
 
 
 
 A
 ∈
 
 
 C
 
 
 m
 ×
 n
 
 
 
 
 {\displaystyle A\in \mathbb {C} ^{m\times n}}
 
 by requiring 
 
 
 
 U
 ∈
 
 
 C
 
 
 m
 ×
 n
 
 
 
 
 {\displaystyle U\in \mathbb {C} ^{m\times n}}
 
 to be a <a href="/facts/Semi-orthogonal_matrix/XSQd0jH6">semi-unitary</a> matrix, and 
 
 
 
 P
 ∈
 
 
 C
 
 
 n
 ×
 n
 
 
 
 
 {\displaystyle P\in \mathbb {C} ^{n\times n}}
 
 to be a positive-semidefinite Hermitian matrix. The decomposition always exists, and 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is always unique. The matrix 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is unique if and only if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 has full rank.

Polar decomposition open-in-new

Polar decomposition