{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T19:13:16Z","timestamp":1790795596205,"version":"4.1.0"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T00:00:00Z","timestamp":1287360000000},"content-version":"unspecified","delay-in-days":9209,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1985,8]]},"abstract":"<jats:p>Let <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub> be a square matrix (complex or otherwise) and <jats:italic>u<\/jats:italic><jats:sub>0<\/jats:sub> a (normalized) eigenvector associated with an eigenvalue \u03bb<jats:sup>o<\/jats:sup> of <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>, so that the triple (<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>u<\/jats:italic><jats:sub>0<\/jats:sub>, \u03bb<jats:sub>0<\/jats:sub>) satisfies the equations <jats:italic>Xu<\/jats:italic> = \u03bb<jats:italic>u<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600011129_inline1\"\/>. We investigate the conditions under which unique differentiable functions \u03bb(<jats:italic>X<\/jats:italic>) and <jats:italic>u<\/jats:italic>(<jats:italic>X<\/jats:italic>) exist in a neighborhood of <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub> satisfying \u03bb(<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>) = \u03bb<jats:sub>O<\/jats:sub>, <jats:italic>u<\/jats:italic>(<jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub>) = <jats:italic>u<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>X<jats:sub>u<\/jats:sub><\/jats:italic> = \u03bb<jats:italic>u<\/jats:italic>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600011129_inline1\"\/>. We obtain the first and second derivatives of \u03bb(<jats:italic>X<\/jats:italic>) and the first derivative of <jats:italic>u<\/jats:italic>(<jats:italic>X<\/jats:italic>). Two alternative expressions for the first derivative of \u03bb(<jats:italic>X<\/jats:italic>) are also presented.<\/jats:p>","DOI":"10.1017\/s0266466600011129","type":"journal-article","created":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T05:12:12Z","timestamp":1287378732000},"page":"179-191","source":"Crossref","is-referenced-by-count":186,"title":["On Differentiating Eigenvalues and Eigenvectors"],"prefix":"10.1017","volume":"1","author":[{"given":"Jan R.","family":"Magnus","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2010,10,18]]},"reference":[{"key":"S0266466600011129_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1765(82)90097-0"},{"key":"S0266466600011129_ref001","first-page":"135","article-title":"On the matrix differentiation of the characteristic roots of matrices","volume":"8","author":"Bargmann","year":"1974","journal-title":"South African Statistical Journal"},{"key":"S0266466600011129_ref003","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176344621"},{"key":"S0266466600011129_ref004","doi-asserted-by":"publisher","DOI":"10.1137\/0601049"},{"key":"S0266466600011129_ref005","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9574.1967.tb00550.x"},{"key":"S0266466600011129_ref007","doi-asserted-by":"publisher","DOI":"10.1080\/03610927308827036"},{"key":"S0266466600011129_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386087"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600011129","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T17:59:10Z","timestamp":1558547950000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600011129\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,8]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1985,8]]}},"alternative-id":["S0266466600011129"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600011129","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,8]]}}}