Symmetry decomposition and matrix multiplication

Higham, Nicholas J., Lettington, Matthew C. ORCID: https://orcid.org/0000-0001-9327-143X and Schmidt, Karl Michael ORCID: https://orcid.org/0000-0002-0227-3024 2025. Symmetry decomposition and matrix multiplication. Linear Algebra and its Applications 710 , pp. 310-335. 10.1016/j.laa.2025.01.041

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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 relates 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Publisher:	Elsevier
ISSN:	0024-3795
Funders:	The Royal Society
Date of First Compliant Deposit:	1 February 2025
Date of Acceptance:	31 January 2025
Last Modified:	04 Mar 2025 14:47
URI:	https://orca.cardiff.ac.uk/id/eprint/175819

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