Fast matrix algebra for dense matrices with rank-deficient off-diagonal blocks

All direct solvers described in this text make frequent use of matrix operations such as matrixmatrix multiplications, matrix inversions, LU factorizations, etc. The matrices that are manipulated are almost all dense, but fortunately, they will be either of small size, or will have internal structure that allows the required operations to be performed rapidly even though the matrices are dense. To be precise, the “internal structure” that is exploited is that off-diagonal blocks of the matrices can be well approximated by matrices of low rank. The chapter will illustrate the key ideas by introducing a very simple set of “compressible” matrices, and then showing how to rapidly perform algebraic operations on such compressible matrices.