{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T00:55:32Z","timestamp":1774745732852,"version":"3.50.1"},"reference-count":9,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore\u2013Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore\u2013Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.<\/jats:p>","DOI":"10.3390\/sym13081386","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:46:44Z","timestamp":1627854404000},"page":"1386","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["When Does a Dual Matrix Have a Dual Generalized Inverse?"],"prefix":"10.3390","volume":"13","author":[{"given":"Firdaus E.","family":"Udwadia","sequence":"first","affiliation":[{"name":"Aerospace and Mechanical Engineering, Civil and Environmental Engineering, Information and Operations Management, University of Southern California, Los Angeles, CA 90089, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"ref_1","unstructured":"Fischer, I. (1998). Dual-Number Methods in Kinematics, Statics and Dynamics, CRC Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.mechmachtheory.2013.12.016","article-title":"Dual Tensors Based Solutions for Rigid Body Motion Parameterization","volume":"74","author":"Condurache","year":"2014","journal-title":"Mech. Mach. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1109\/JRA.1987.1087138","article-title":"Dual-number Transformations and its Applications to Robotics","volume":"3","author":"Gu","year":"1987","journal-title":"IEEE J. Robot. Autom."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Udwadia, F.E., Pennestri, E., and de Falco, D. (2020). Do All Dual Matrices have Dual Moore-Penrose Inverses?. Mech. Mach. Theory, 151.","DOI":"10.1016\/j.mechmachtheory.2020.103878"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Udwadia, F.E. (2021). Dual Generalized Inverses and their Use in Solving Systems of Linear Dual Equations. Mech. Mach. Theory, 156.","DOI":"10.1016\/j.mechmachtheory.2020.104158"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yanai, H., Takeuchi, K., and Takane, Y. (2011). Projection Matrices, Generalized Inverse Matrices, and Singular Values, Springer.","DOI":"10.1007\/978-1-4419-9887-3"},{"key":"ref_7","unstructured":"Udwadia, F.E., and Kalaba, R.E. (2007). Analytical Dynamics: A New Approach, Cambridge University Press."},{"key":"ref_8","unstructured":"Rao, C.R., and Mitra, S.K. (1972). Generalized Inverses and Its Applications, John Wiley."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1017\/S0305004100030401","article-title":"A Generalized Inverse for Matrices","volume":"51","author":"Penrose","year":"1955","journal-title":"Math. Proc. Camb. Philos. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1386\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:37:15Z","timestamp":1760164635000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,30]]},"references-count":9,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081386"],"URL":"https:\/\/doi.org\/10.3390\/sym13081386","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,30]]}}}