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 low rank submatrices starting from the bottom left matrix corner. This representation will be compared to the (block) quasiseparable representations occurring in the literature. We will then provide some basic algorithms for the Givens-weight representation, in particular showing how to obtain a Givens-weight representation for a full matrix, and how to reduce the order of the representation, whenever appropriate. We will also show how to update the representation under the action of Givens transformations, and how to compute matrix-vector products. As such, these results will be the basis for the algorithms on Givens-weight representations to be described in subsequent papers.