  This paper considers the foundational question of the existence of a
fundamental (resp. essential) matrix given $m$ point correspondences in two
views. We present a complete answer for the existence of fundamental matrices
for any value of $m$. Using examples we disprove the widely held beliefs that
fundamental matrices always exist whenever $m \leq 7$. At the same time, we
prove that they exist unconditionally when $m \leq 5$. Under a mild genericity
condition, we show that an essential matrix always exists when $m \leq 4$. We
also characterize the six and seven point configurations in two views for which
all matrices satisfying the epipolar constraint have rank at most one.
