A Givens-Weight Representation for Rank Structured Matrices

In this paper we introduce a Givens-weight representation for rank structured matrices, where the rank structure is defined by certain submatrices starting from the bottom left or upper right matrix corner being of low rank. We proceed in two steps. First, we introduce a unitary-weight representation. This representation will be compared to the (block) quasiseparable representations introduced by P. Dewilde and A.-J. van der Veen [Time-varying Systems and Computations, Kluwer Academic Publishers, Boston, 1998]. More specifically, we show that our unitary-weight representations are theoretically equivalent to the so-called block quasiseparable representations in input or output normal form introduced by Dewilde and van der Veen [Time varying Systems and Computations, Kluwer Academic Publishers, Boston, 1998]. Next, we move from the unitary-weight to the Givens-weight representation. We then provide some basic algorithms for the unitary/Givens-weight representation, showing how to obtain such a representation for a dense matrix by means of numerical approximation. We also show how to “swap” the representation and how to reduce the number of parameters of the representation, whenever appropriate. As such, these results will become the basis for algorithms on unitary/Givens-weight representations to be described in subsequent papers.