  We consider height functions on symmetric spaces $M\cong G/K$ embedded in the
associated matrix Lie group $G$. In particular we study the relationship
between the critical sets of the height function on $G$ and its restriction to
$M$. Also we prove that the gradient flow on $M$ can be integrated by means of
a generalized Cayley transform. This allows to obtain explicit local charts for
the critical submanifolds. Finally, we discuss how to reduce the generic case
to a height function whose ground hyperplane is orhogonal to a real diagonal
matrix. This result requires to prove the existence of a polar decomposition
adapted to the automorphism defining $M$. Detailed examples are given.
