  The local Euler obstructions and the Euler characteristics of linear sections
with all hyperplanes on a stratified projective variety are key geometric
invariants in the study of singularity theory. Despite their importance, in
general it is very hard to compute them. In this paper we consider a special
type of singularity: the recursive group orbits. They are the group orbits of a
sequence of $G_n$ representations $V_n$ satisfying certain assumptions. We
introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it
to give explicit formulas to the local Euler obstructions and the sectional
Euler characteristics of such orbits. In particular, the matrix rank loci are
examples of recursive group orbits. Thus as applications, we explicitly compute
these geometry invariants for ordinary, skew-symmetric and symmetric rank loci.
Our method is systematic and algebraic, thus works for algebraically closed
field of characteristic $0$. Moreover, in the complex setting we also compute
the stalk Euler characteristics of the Intersection Cohomology Sheaf complexes
for all three types of rank loci.
