  We introduce homing vector automata, which are finite automata augmented by a
vector that is multiplied at each step by a matrix determined by the current
transition, and have to return the vector to its original setting in order to
accept the input. The computational power and properties of deterministic,
nondeterministic, blind, non-blind, real-time and one-way versions of these
machines are examined and compared to various related types of automata. A
generalized version of the Stern-Brocot encoding method, suitable for
representing strings on arbitrary alphabets, is also developed.
