In an earlier post (Use Lie Sub-Groups of GL(3, R) for elastic deformation ? [here](https://mathoverflow.net/questions/40470/use-lie-sub-groups-of-gl3-r-for-elastic-deformation?)), I mentioned polar decompositions as in F = RU where R in SO(3) & U in symmetric positive-semidefinite matrices. In response, I received the following comment: "The decompositions you mention are well-known in the theory of Lie groups, e.g., F=RU is called Iwasawa decomposition." 

I have been trying to understand how a polar decomposition & Iwasawa decomposition are related. I have found the following definitions of both.
Iwasawa: G=KAN for G in GL(n, R) where K = orthogonal matrices (rot'ns ?), A = positive diag. matrices & N = upper triag. matrices with diag. entries = 1. 
Polar: G=KAK where G = semi-simple Lie group, K = maximal compact subgroup of G, A = abelian subgroup of G. For GL(n, R), I think K in SO(n) but unsure about A. The only abelian subgroup of GL(n, R) appears to be the set of all nonzero scalar matrices (i.e. scalar multiples of identity matrix). I'm also puzzled by G=KAK having 3 terms but F=RU only 2. 

Given all of this, I'm still unsure how these two decompositions are related.