In recent years, fast direct integral equation solvers have been developed, as an alternative to iterative solver, for problems that suffer from ill-conditioning or when a solution is sought for many right-hand-sides. These solvers rely on the fast computation of a compressed representation of the impedance matrix. Then, the compressed representation's factorized (effectively 'solved') form is computed and applied to each right-hand-side. If the factorized form inherits the original matrix's compressibility, the savings in memory are maintained and the solution for each right-hand-side is, indeed, fast. The compression of the impedance matrix is often performed in a hierarchical manner. The geometry is first partitioned into clusters of basis and testing functions. Then, a hierarchical block structure for compression is defined, in accordance with the choice of hierarchical algebraic procedure for computing the compressed factorized form, e.g., , , or . Matrix blocks, corresponding to interactions between sources and observers, that are assumed compressible, in some sense, are identified and compressed.