Work

Symmetry decomposition and matrix multiplication

Year: 2025

Type: article

Abstract: General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices , the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relate to the pseudoinverses of its component parts. For invertible matrices , this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.

Source: Linear Algebra and its Applications

DOI: 10.1016/j.laa.2025.01.041

Authors: Nicholas J. Higham, Matthew C. Lettington, Karl Michael Schmidt

Institutions: University of Manchester, Cardiff University

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,230

Funder: Royal Society