Relation To Polar Decomposition
Consider with the Cartan involution . Then is the real Lie algebra of skew-symmetric matrices, so that, while is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. Notice that the polar decomposition of an invertible matrix is unique.
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