  In graph theory a partition of the vertex set of a graph is called equitable
if for all pairs of cells all vertices in one cell have an equal number of
neighbours in the other cell. Considering the implications for the adjacency
matrix one may generalize that concept as a block partition of a complex square
matrix s.t. each block has constant row sum. It is well known that replacing
each block by its row sum yields a smaller matrix whose multiset of eigenvalues
is contained in the initial spectrum. We generalize this approach to weighted
row sums and rectangular matrices and derive an efficient unitary
transformation which approximately block triangularizes a matrix w.r.t. an
arbitrary partition. Singular values and Hermiticity (if present) are
preserved. The approximation is exact in the equitable case and the error can
be bounded in terms of unitarily invariant matrix norms.

Discrete Mathematics