| 1. | Notice that the polar decomposition of an invertible matrix is unique.
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| 2. | Indeed, they each arise in polar decomposition of a complex number.
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| 3. | AW *-algebras also always have polar decomposition.
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| 4. | Just as the classic polar decomposition, the DPD is valid in any finite dimension.
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| 5. | Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition.
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| 6. | Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix.
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| 7. | To conserve momentum the rotation of the body must be estimated properly, for example via polar decomposition.
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| 8. | The resulting unitary operator " U " then yields the polar decomposition of " A ".
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| 9. | An appeal to polar decomposition extend this to the general case where " T f " need not be positive.
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| 10. | Furthermore, equality is achieved if " U " * is the unitary operator in the polar decomposition of " A ".
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