Polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle P} is a positive semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix, and P {\displaystyle P} is a positive semi-definite symmetric matrix 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 linear transformation of n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} , the polar decomposition separates it into a rotation or reflection U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} and a scaling of the space along a set of n {\displaystyle n} orthogonal axes.