Rank-r Matrices

Rank-r matrices will be an important stepping stone towards hierarchical matrices. Since typically we assume r to be of moderate size, also the term low-rank matrices is used. The storage of rank-r matrices as well as operations involving rank-r matrices form the basis of the hierarchical matrix representation and the hierarchical matrix operations, since these are reduced to additions and multiplications by rank-r or small full matrices.In Section 2.1 we recall the rank of a matrix and various properties related to the matrix rank. As shown in Section 2.2, rank-r matrices allow for a suitable representation with low storage cost. We introduce the notation $$\mathcal{R}(r, I, J)$$ for matrices presented in this format. To avoid a possible misunderstanding, we note that rank-r matrices need not have the exact rank r, but are at most of rank r. In Section 2.3 we discuss the arithmetical work of matrix-vector multiplication, matrix-matrix addition, and matrix-matrix multiplication by rank-r matrices. In Section 2.4 we recall that the best approximation of a general matrix by a rank-r matrix makes use of the singular value decomposition (SVD) (details and proofs in Appendix C.2). Approximating a rank-s matrix by another matrix of smaller rank r < s, as discussed in Section 2.5, will become an important tool. The QR decomposition and the reduced QR decomposition are defined. In Section 2.6 we apply tools of the previous subsections and introduce formatted addition involving the characteristic truncation to the low-rank format. In Section 2.7 we mention a modification of the standard representation $$\mathcal{R}(r, I, J)$$ .