rdfs:comment
  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 positivesemidefinite Hermitian matrix, 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 can also be defined as where and have the same properties as above (but are different matrices, in general, for the same ).
 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 positivesemidefinite Hermitian matrix, 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.

has abstract
  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 positivesemidefinite Hermitian matrix, 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 positivedefinite. In that case, can be written uniquely in the form , where is unitary and is the unique selfadjoint 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 and have the same properties as above (but are different matrices, in general, for the same ). 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 nonnegative real number), and is a complex number with unit norm (an element of the circle group).
 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 positivesemidefinite Hermitian matrix, 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 positivedefinite. In that case, can be written uniquely in the form , where is unitary and is the unique selfadjoint 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 symmetric positivedefinite but is in general a different matrix, while is the same matrix as above. 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 nonnegative real number), and is a complex number with unit norm (an element of the circle group).
 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 positivesemidefinite Hermitian matrix, 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 positivedefinite. In that case, can be written uniquely in the form , where is unitary and is the unique selfadjoint 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 symmetric positivedefinite but is in general a different matrix, while is the same matrix as above. 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 nonnegative 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 semiunitary matrix and to be a positivesemidefinite Hermitian matrix. The decomposition always exists and is always unique. The matrix is unique if and only if has full rank.
