  New contributions are offered to the theory and practice of the Discrete
Empirical Interpolation Method (DEIM). These include a detailed
characterization of the canonical structure; a substantial tightening of the
error bound for the DEIM oblique projection, based on index selection via a
strong rank revealing QR factorization; and an extension of the DEIM
approximation to weighted inner products defined by a real symmetric
positive-definite matrix $W$. The weighted DEIM ($W$-DEIM) can be deployed in
the more general framework where the POD Galerkin projection is formulated in a
discretization of a suitable energy inner product such that the Galerkin
projection preserves important physical properties such as e.g. stability.
Also, a special case of $W$-DEIM is introduced, which is DGEIM, a discrete
version of the Generalized Empirical Interpolation Method that allows
generalization of the interpolation via a dictionary of linear functionals.
