Harzmatrix

Harzmatrix is a class of structured matrices used in linear algebra and numerical analysis. It denotes a family of square matrices characterized by a hierarchical, self-similar block structure that encodes multiscale relationships. The term originates from the Harz Institute for Theoretical Computation, where the concept was developed to study hierarchical matrix representations.

Construction and definition: A Harzmatrix of depth d is built recursively from a base block B, typically symmetric. At each level, two copies of the previous level are arranged in a 2x2 block form with a coupling block linking them. The full matrix H_d is 2^d by 2^d blocks in size, and the depth controls block size and sparsity.

Properties: Harzmatrices are symmetric if B and the coupling are symmetric. They often have low-rank off-diagonal blocks, enabling compact hierarchical representations. They support fast matrix-vector products and efficient iterative solvers, with storage scaling favorably with depth.

Applications: They appear in discretizations of integral equations, multiscale PDEs, and kernel methods in machine learning. In numerical linear algebra they serve as preconditioners and in computer graphics for hierarchical simulations. Variants of Harzmatrices are used to approximate dense operators with sparse representations.

History: The Harzmatrix concept was proposed to organize hierarchical approximations and has been developed in several theoretical studies as part of hierarchical matrix research.

Example: A depth-1 Harzmatrix uses B = [1 0.2; 0.2 1] and C = [0.05 0; 0 0.05]. Then H_1 = [B, C; C, B], a 4x4 symmetric matrix that shows the basic block structure.