  In the present work, we demonstrate how the pseudoinverse concept from linear
algebra can be used to represent and analyze the boundary conditions of linear
systems of partial differential equations. This approach has theoretical and
practical implications; the theory applies even if the boundary operator is
rank deficient, or near rank deficient. If desired, the pseudoinverse can be
implemented directly using standard tools like Matlab. We also introduce a new
and simplified version of the semidiscrete approximation of the linear PDE
system, which completely avoids taking the time derivative of the boundary
data. The stability results are valid for general, nondiagonal
summation-by-parts norms. Another key result is the extension of
summation-by-parts operators to multi-domains by means of carefully crafted
embedding operators. No extra numerical boundary conditions are required at the
grid interfaces. The aforementioned pseudoinverse allows for a compact
representation of these multi-block operators, which preserves all relevant
properties of the single-block operators. The embedding operators can be
constructed for multiple space dimensions. Numerical results for the
two-dimensional Maxwell's equations are presented, and they show very good
agreement with theory.
