Unlike the classic polar decomposition, however, the dynamic rotation and stretch tensors are obtained from solving linear differential equations, rather than from matrix manipulations.
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*PM : polar decomposition in von Neumann algebras, id = 9868 new !-- WP guess : polar decomposition in von Neumann algebras-- Status:
13.
*PM : polar decomposition in von Neumann algebras, id = 9868 new !-- WP guess : polar decomposition in von Neumann algebras-- Status:
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Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition.
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Via polar decomposition, one can prove that the space of " p " th Schatten class operators is an ideal in " B ( H ) ".
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This is because that in the finite dimensional case, the partial isometry " U " in the polar decomposition " A " = " UP " can be taken to be unitary.
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In the same way as a complex number can be represented in a polar form, one has a " polar decomposition " for a complex measure : There exists a measurable function & theta; with real values such that
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The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of the infinitesimal operator corresponding to an element in the maximal cone.
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Also, the closure of the image under an operator of " M " of any subspace belonging to " M " also belongs to " M " . ( These results are a consequence of the polar decomposition ).
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The deformation gradient \ mathbf { F } \, \ !, like any second-order tensor, can be decomposed, using the polar decomposition theorem, into a product of two second-order tensors ( Truesdell and Noll, 1965 ) : an orthogonal tensor and a positive definite symmetric tensor, i . e.
