Structure and approximation properties of Laplacian-like matrices

TL;DR

This paper characterizes Laplacian-like matrices, explores their algebraic properties, and proposes an algorithm for projecting matrices onto this subspace, enhancing techniques for solving large linear systems efficiently.

Contribution

It provides a detailed characterization of Laplacian-like matrices and introduces an explicit projection algorithm, advancing the understanding and computational handling of these matrices.

Findings

Laplacian-like matrices form a Lie sub-algebra.

The paper characterizes key properties of these matrices.

An algorithm for orthogonal projection onto the subspace is proposed.

Abstract

Many of today's problems require techniques that involve the solution of arbitrarily large systems $A\mathbf{x}=\mathbf{b}$. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix $A \in \mathbb{R}^{N\times N}.$