The Polar Decomposition Theorem

The polar decomposition theorem expresses a general second order tensor as a product of a positive symmetric tensor with an orthogonal tensor. This is very useful in interpreting deformation processes in terms of a translation, a rigid rotation of the principal axes of strain, and stretching along these axes (Jaunzemis 1967). 

Let F be a nonsingular asymmetric second-order tensor. Note that the deformation gradient tensor F happens to be an asymmetric tensor. Then, F allows the unique representations 

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Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

Another approach to the analysis of Jones and Mueller-Jones matrix exploits the polar decomposition theorem [18]. This approach was first suggested in [19] and was explored in [20,21]. The polar decomposition of a Jones matrix J can be represented as ... 

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