Polar Factorization of a Matrix

It is known that if A is a bounded linear operator with closed range on a Hilbe rt space then A can be fac tored as A = UH, with U a partial isometry and H nonnegative and self adjoint. For the finite dimensional case a s tri ctly matrix-theoretic derivation is given based on the concept of a ge neralized inverse. Certain properti es of the factors are give n as well as conditions under whic h H or both U and H are uniquely de termined by A. A pivotal ite m in the derivation is the representation of a square partial isometry a s the produc t of a unitary matrix and a n orthogonal projection. Thi s representa tion is new, of some int e rest in itse lf and greatl y s impli fies the de rivations.