In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is an orthogonal matrix and is a positive semi-definite symmetric matrix ( is a unitary matrix and is a positive semi-definite Hermitian matrix in the complex 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. Intuitive interpretation 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 single rotation or reflection of (the action of ). Alternatively, the decomposition expresses the transformation defined by as a rotation () followed by a scaling () along certain orthogonal directions. The scale factors are the same, but the directions are different. Properties The polar decomposition of the complex conjugate of is given by Note thatgives the corresponding polar decomposition of the determinant of A, since and . In particular, if