  We introduce and study isomonodromy transformations of the matrix linear
difference equation Y(z+1)=A(z)Y(z) with polynomial (or rational) A(z). Our
main result is a construction of an isomonodromy action of Z^{m(n+1)-1} on the
space of coefficients A(z) (here m is the size of matrices and n is the degree
of A(z)). The (birational) action of certain rank n subgroups can be described
by difference analogs of the classical Schlesinger equations, and we prove that
for generic initial conditions these difference Schlesinger equations have a
unique solution. We also show that both the classical Schlesinger equations and
the Schlesinger transformations known in the isomonodromy theory, can be
obtained as limits of our action in two different limit regimes.
  Similarly to the continuous case, for m=n=2 the difference Schlesinger
equations and their q-analogs yield discrete Painleve equations; examples
include dPII, dPIV, dPV, and q-PVI.
