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: | 13 Jun 2025 15:23 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/175819 |
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