Support theory is a methodology for bounding eigenvalues and generalized eigenvalues of matrices and matrix pencils; such bounds have been stated both in algebraic terms and in combinatorial terms based on embeddings of the underlying graphs of the matrices. In this paper, we present a theorem that demonstrates the connection between these various bounding techniques and also suggest a possible approach to generating approximate inverses for preconditioning. The theorem shows, given matrices $A = U D_A U^*$ and $B = V D_B V^*$ (where $D_A$ and $D_B$ are invertible Hermitian matrices, and $U$ and $V$ are not necessarily square), that it is possible to define a matrix $W$ such that $W^* D_B^{-1} W D_A$ has the same nonzero eigenvalues counting multiplicity as $B^+ A$. In the special case that $U$ is the orthogonal projector onto the range space of $B$ and $D_A = I$ (and hence that $A = U U^* = U^2 = U$), then $W^* D_B^{-1} W = B^+$. This suggests that finding an approximation to $W$ might lead to an approximate inverse that can be used in preconditioning. We also describe how this theorem generalizes the idea of graph embeddings in an algebraic sense.