  We consider $m \times s$ matrices (with $m\geq s$) in a real affine subspace
of dimension $n$. The problem of finding elements of low rank in such spaces
finds many applications in information and systems theory, where low rank is
synonymous of structure and parsimony. We design computer algebra algorithms,
based on advanced methods for polynomial system solving, to solve this problem
efficiently and exactly: the input are the rational coefficients of the
matrices spanning the affine subspace as well as the expected maximum rank, and
the output is a rational parametrization encoding a finite set of points that
intersects each connected component of the low rank real algebraic set. The
complexity of our algorithm is studied thoroughly. It is polynomial in
$\binom{n+m(s-r)}{n}$. It improves on the state-of-the-art in computer algebra
and effective real algebraic geometry. Moreover, computer experiments show the
practical efficiency of our approach.
