Displacement operators relative to group matrices

Structured Matrices and the Algebra of Displacement Operators

Matrix calculations underlie countless problems in science, mathematics, and engineering. When the involved matrices are highly structured, displacement operators can be used to accelerate fundamental operations such as matrix-vector multiplication. In this thesis, we provide an introduction to the theory of displacement operators and study the interplay between displacement and natural matrix ...

On matrices with displacement structure: generalized operators and faster algorithms

For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product AB, where A is a structured n × n matrix of displacement rank α, and B is an arbitrary n × α matrix. Given B and a so-called generator of A, this product is classically computed with a cost ranging from O(α...

Linear Operators on Matrices: Preserving Spectrum and Displacement Structure

In this paper we characterize those linear operators on general matrices that preserve singular values and displacement rank. We also characterize those linear operators on Hermitian matrices that preserve eigenvalues and displacement inertia.

Inverse Displacement Operators

Inversion of Displacement Operators

We recall briefly the displacement rank approach to the computations with structured matrices, which we trace back to the seminal paper by Kailath, Kung, and Morf [J. Math. Anal. Appl., 68 (1979), pp. 395–407]. The concluding stage of the computations is the recovery of the output from its compressed representation via the associated displacement operator L. The recovery amounts to the inversio...