  Efficient computer implementations of the GW approximation must approximate a
numerically challenging frequency integral; the integral can be performed
analytically, but doing so leads to an expensive implementation whose
computational cost scales as $O(N^6)$ where $N$ is the size of the system. Here
we introduce a new formulation of the full-frequency GW approximation by
exactly recasting it as an eigenvalue problem in an expanded space. This new
formulation (1) avoids the use of time or frequency grids, (2) naturally
precludes the common "diagonal" approximation, (3) enables common iterative
eigensolvers that reduce the canonical scaling to $O(N^5)$, and (4) enables a
density-fitted implementation that reduces the scaling to $O(N^4)$. We
numerically verify these scaling behaviors and test a variety of approximations
that are motivated by this new formulation. In this new formulation, the
relation of the GW approximation to configuration interaction, coupled-cluster
theory, and the algebraic diagrammatic construction is made especially
apparent, providing a new direction for improvements to the GW approximation.
