  We explore various aspects of implementing the full M-theory U-duality group
E_{d+1}, and thus Lorentz invariance, in the finite N matrix theory (DLCQ of
M-theory) on d-tori: (1) We generalize the analysis of U-duality orbits of BPS
states by Elitzur et al. (hep-th/9707217) from E_{d} to E_{d+1}. (2) We
identify the new E_{d+1}-symmetries with Nahm-duality-like symmetries
(N-duality) exchanging the rank N of the matrix theory gauge group with other
quantum numbers. (3) We describe the action of N-duality on BPS bound states,
thus making testable predictions for the Lorentz invariance of matrix theory.
(4) We discuss the problems that arise in the matrix theory limit for BPS
states with no top-dimensional branes, i.e. configurations with N=0. (5) We
show that N-duality maps the matrix theory SYM picture to the matrix string
picture and argue that, for d even, the latter should be thought of as an
M-theory membrane description (which appears to be well defined even for d>5).
(6) We find a compact and unified expression for a U-duality invariant of
E_{d+1} for all d and show that in d=5,6 it reduces to the black hole entropy
cubic E_{6}- and quartic E_{7}-invariants respectively. (7) We describe some of
the solitonic states in d=6,7 and give an example (a `rolled-up' Taub-NUT
6-brane) of a configuration exhibiting the unusual 1/g_{s}^{3}-behaviour.
