  The matrix product representation provides a useful formalism to study not
only entangled states, but also entangled operators in one dimension. In this
paper, we focus on unitary transformations and show that matrix product
operators that are unitary provides a necessary and sufficient representation
of 1D unitaries that preserve locality. That is, we show that matrix product
operators that are unitary are guaranteed to preserve locality by mapping local
operators to local operators while at the same time all locality preserving
unitaries can be represented in a matrix product way. Moreover, we show that
the matrix product representation gives a straight-forward way to extract the
GNVW index defined in Ref.\cite{Gross2012} for classifying 1D locality
preserving unitaries. The key to our discussion is a set of `fixed point'
conditions which characterize the form of the matrix product unitary operators
after blocking sites. Finally, we show that if the unitary condition is relaxed
and only required for certain system sizes, the matrix product operator
formalism allows more possibilities than locality preserving unitaries. In
particular, we give an example of a simple matrix product operator which is
unitary only for odd system sizes, does not preserve locality and carries a
`fractional' index as compared to their locality preserving counterparts.
