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Concept
Polar decomposition
Summary
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
The polar decomposition of a square matrix always exists. If is invertible, the decomposition is unique, and the factor will be positive-definite. In that case, can be written uniquely in the form , where is unitary and is the unique self-adjoint logarithm of the matrix . This decomposition is useful in computing the fundamental group of (matrix) Lie groups.
The polar decomposition can also be defined as where is a symmetric positive-definite matrix with the same eigenvalues as but different eigenvectors.
The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number as , where is its absolute value (a non-negative real number), and is a complex number with unit norm (an element of the circle group).
The definition may be extended to rectangular matrices by requiring to be a semi-unitary matrix and to be a positive-semidefinite Hermitian matrix. The decomposition always exists and is always unique. The matrix is unique if and only if has full rank.
A real square matrix can be interpreted as the linear transformation of that takes a column vector to . Then, in the polar decomposition , the factor is an real orthonormal matrix. The polar decomposition then can be seen as expressing the linear transformation defined by into a scaling of the space along each eigenvector of by a scale factor (the action of ), followed by a rotation of (the action of ).
Alternatively, the decomposition expresses the transformation defined by as a rotation () followed by a scaling () along certain orthogonal directions.
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