{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:23:09Z","timestamp":1760145789614,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T00:00:00Z","timestamp":1724371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Fund Project of Guangxi Minzu University","award":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"],"award-info":[{"award-number":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"]}]},{"name":"Guangxi Science and Technology Department Specific Research Project of Guangxi for Research Bases and Talents","award":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"],"award-info":[{"award-number":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"]}]},{"name":"Thousands of Young and Middle-aged Key Teachers Training Programme in Guangxi Colleges and Universities","award":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"],"award-info":[{"award-number":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"]}]},{"name":"Joint Research Project between China and Serbia","award":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"],"award-info":[{"award-number":["2019KJQD03","GHIKE-AD23023001","GUIJIAOSHIFAN2019-81HAO","2024-6-7"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A^, (which is usually represented in the form A^=A+\u03b5A0, A and A0 are, respectively, the standard part and the infinitesimal part of A^) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0.<\/jats:p>","DOI":"10.3390\/sym16091103","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:58:07Z","timestamp":1724417887000},"page":"1103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Perturbation of Dual Group Generalized Inverse and Group Inverse"],"prefix":"10.3390","volume":"16","author":[{"given":"Tianhe","family":"Jiang","sequence":"first","affiliation":[{"name":"Office of Computer and Mathematics, Department of Public Infrastructure, Guangxi Health Science College, Nanning 530023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2569-0821","authenticated-orcid":false,"given":"Hongxing","family":"Wang","sequence":"additional","affiliation":[{"name":"Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6192-0546","authenticated-orcid":false,"given":"Yimin","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. 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