  Let $G=GL_{n}(R)$, with the usual Cartan decomposition $G=Kexp(p_0)$ and the
usual Iwasawa decomposition $G=NAK$. We determine the image of $exp(p_0)$ under
the projection of $G$ to $K$ through the Iwasawa decomposition. As an
application, we prove a positivity result about the matrix coefficients of
principal series representations of $G$.
