{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:51:36Z","timestamp":1774630296229,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Using the Dunford\u2013Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy\u2019s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica\u00a9, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.<\/jats:p>","DOI":"10.3390\/axioms11020051","type":"journal-article","created":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T21:59:55Z","timestamp":1643320795000},"page":"51","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Logarithm of a Non-Singular Complex Matrix via the Dunford\u2013Taylor Integral"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0969-884X","authenticated-orcid":false,"given":"Diego","family":"Caratelli","sequence":"first","affiliation":[{"name":"Department of Research and Development, The Antenna Company, High Tech Campus 29, 5656 AE Eindhoven, The Netherlands"},{"name":"Electromagnetics Group, Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7899-3087","authenticated-orcid":false,"given":"Paolo Emilio","family":"Ricci","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, International Telematic University UniNettuno, 39 Corso Vittorio Emanuele II, I-00186 Rome, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,27]]},"reference":[{"key":"ref_1","unstructured":"Gantmacher, F.R. (1959). The Theory of Matrices, Chelsea Pub. Co."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Higham, N.J. (2008). Functions of Matrices: Theory and Computation, SIAM.","DOI":"10.1137\/1.9780898717778"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"C153","DOI":"10.1137\/110852553","article-title":"Improved inverse scaling and squaring algorithms for the matrix logarithm","volume":"34","author":"Higham","year":"2012","journal-title":"SIAM J. Sci. 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