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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-semidefinite 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 ).

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  • 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-semidefinite 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 positive-semidefinite 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.
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  • Polar decomposition
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  • 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-semidefinite 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 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 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 non-negative 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 positive-semidefinite 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 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 symmetric positive-definite 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 non-negative 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 positive-semidefinite 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 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 symmetric positive-definite 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 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.
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