{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:16:42Z","timestamp":1787379402938,"version":"build-2736575974"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Let $P : \\Omega \\subset {\\mathbb C} \\rightarrow {\\mathbb C}^{n\\times n}$ be given by $P(\\lambda) :=\\sum^m_{j=0}A_j\\phi_j(\\lambda),$ where $ \\phi_j : \\Omega \\rightarrow {\\mathbb C}$ for $j=0, 1, \\ldots, m$ are suitable functions. We present an eigenvector-free framework for the sensitivity analysis of eigenvalues of $P.$ We analyze the Fr\u00e9chet differentiability of a simple eigenvalue of $P$ as a function of $P$ and derive two equivalent representations of the Fr\u00e9chet derivative and the gradient of the eigenvalue. Further, we derive three equivalent representations of the condition number $\\mathrm{cond}(\\lambda, P)$ of a simple eigenvalue $\\lambda$ of $P.$ Specially, we present an eigenvector-free representation of $\\mathrm{cond}(\\lambda, P)$ which generalizes a result due to Smith [ Numer. Math., 10 (1967), pp. 232--240] for a standard eigenvalue problem to the case of a nonlinear eigenvalue problem and provides an alternative viewpoint of the sensitivity of eigenvalues. In the second part, we consider a homogeneous matrix-valued function $H : {\\mathbb C}^2 \\rightarrow {\\mathbb C}^{n\\times n}$ of the form $H(c, s) :=\\sum^m_{j=0}A_j\\psi_j(c, s),$ where $ \\psi_j : {\\mathbb C}^2 \\rightarrow {\\mathbb C}$ for $ j= 0, 1, \\ldots, m$ are homogeneous functions of degree $\\ell.$ We present a simple and concise eigenvector-free framework for the sensitivity analysis of eigenvalues of $H$ that avoids the apparatus of projective spaces. We analyze Fr\u00e9chet differentiability of a simple eigenvalue of $H$ as a function of $H$ and derive two equivalent representations of the Fr\u00e9chet derivative and the gradient of the eigenvalue. Furthermore, we derive three equivalent representations of the condition number $\\mathrm{cond}((\\lambda, \\mu), H)$ of a simple eigenvalue $(\\lambda, \\mu)$ of $H.$ Our eigenvector-free representation of $\\mathrm{cond}((\\lambda, \\mu), H)$ generalizes Smith's eigenvector-free representation of the condition number of a simple eigenvalue of a matrix to the case of a homogeneous nonlinear eigenproblem.<\/jats:p>","DOI":"10.1137\/17m1153236","type":"journal-article","created":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T11:38:33Z","timestamp":1559216313000},"page":"672-695","source":"Crossref","is-referenced-by-count":3,"title":["Sensitivity Analysis of Nonlinear Eigenproblems"],"prefix":"10.1137","volume":"40","author":[{"given":"Rafikul","family":"Alam","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sk.","family":"Safique Ahmad","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,5,30]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.2973"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1008622"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/070711645"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.04.020"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/0614061"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"H. Baumgartel,\n                      Analytic Perturbation Theory for Matrices and Operators\n                      , Birkh\u00e4user-Verlag, Basel, Switzerland, 1985.","DOI":"10.1515\/9783112721810"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1145\/2427023.2427024"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(89)90078-9"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0727079"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01400115"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00366-7"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00423-2"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492917000034"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/050628283"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.10.026"},{"key":"atypb16","unstructured":"T. Kato,\n                      Perturbation Theory for Linear Operators\n                      , Springer-Verlag, New York, 1980."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/050628362"},{"key":"atypb18","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/2190-5983-1-10","volume":"1","author":"Mehrmann V.","year":"2011","journal-title":"J. Math. Ind."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201490007"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(82)90055-X"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/0703023"},{"key":"atypb22","unstructured":"F. Rellich,\n                      Perturbation Theory of Eigenvalue Problems,\n                      Gordon and Breach, New York, 1969."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02162166"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2005.03.034"},{"key":"atypb25","unstructured":"G. W. Stewart and J. Sun,\n                      Matrix Perturbation Theory\n                      , Academic Press, New York, 1990."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/090777542"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1993.1002"},{"key":"atypb28","first-page":"351","volume":"3","author":"Sun J. G.","year":"1985","journal-title":"J. Comput. Math."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(90)90129-Z"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385508"},{"key":"atypb31","first-page":"93","volume":"16","author":"Voss H.","year":"2003","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.01.034"},{"key":"atypb33","unstructured":"J. H. Wilkinson,\n                      The Algebraic Eigenvalue Problem\n                      , Oxford University Press, Oxford, 1965."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/BF01402528"},{"key":"atypb35","first-page":"5","volume":"25","author":"Wilkinson J. H.","year":"1984","journal-title":"Util. Math."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90407-2"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90209-7"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1153236","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:14:17Z","timestamp":1787328857000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1153236"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1153236"],"URL":"https:\/\/doi.org\/10.1137\/17m1153236","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}