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In (F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2c ase,Linear Algebra Appl., 38:239-249, 1981), the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig's formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with rank at least n  1.

Explicit polar decomposition of complex matrices