Two mathematical sources worth consulting are Helgason Differential Geometry, Lie Groups, and Symmetric Spaces (Chapter IX.1) and a paper by Kostant available online from www.numdam.org:
MR0364552 (51 #806) 22E45, Kostant, Bertram, On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 413–455 (1974).

Both of these older sources put the various matrix decompositions into the framework of Lie groups and Lie algebras over $\mathbb{C}$. There are certainly other discussions in the literature which might be helpful but don't come to mind immediately. Anyway, Helgason's viewpoint emphasizes the passage to the symmetric space $G/K$ generalizing the space of symmetric matrices, while Kostant emphasizes decompositions within a Lie group.  .

Two mathematical sources worth consulting are Helgason Differential Geometry, Lie Groups, and Symmetric Spaces (Chapter IX.1) and a paper by Kostant available online from www.numdam.org here; reviewed in MR0364552 (51 #806) 22E45, Kostant, Bertram, On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. Ecole Norm. Sup. (4) 6 (1973), 413–455 (1974).

Both of these older sources put the various matrix decompositions into the framework of Lie groups and the complexified Lie algebras. There are certainly other discussions in the literature which might be helpful but don't come to mind immediately. Anyway, Helgason's viewpoint emphasizes the passage to the symmetric space $G/K$ generalizing the space of symmetric matrices, while Kostant emphasizes decompositions within a Lie group.

ADDED: In his answer Faisal points out more explicitly some of these connections with matrix decompositions. There is a lot of interplay among the Cartan, Iwasawa, and Bruhat decompositions, as shown in the above references. The polar decomposition is part of this story, though in a way it's less refined. All of the viewpoints here are important, but historically the Cartan decomposition has some primacy.