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Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Let $\\mathcal A = \\{A_{ij} \\}_{i, j \\in \\mathcal I}$, where $\\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \\times n_j$. For each $i \\in \\mathcal I$, let $\\mathcal V_i$ be an $n_i$-dimensional vector space. We say $\\mathcal A$ is reducible in the coupled sense if there exist subspaces, $\\mathcal U_i \\subseteq \\mathcal V_i$, with $\\mathcal U_i \\neq \\{0\\}$ for at least one $i \\in \\mathcal I$, and $\\mathcal U_i \\neq \\mathcal V_i$ for at least one $i$, such that $A_{ij} (\\mathcal U_j) \\subseteq \\mathcal U_i$ for all $i, j$. Let $\\mathcal B = \\{B_{ij} \\}_{i, j \\in \\mathcal I}$ also be a doubly indexed family of matrices, where $B_{ij}$ is $m_i \\times m_j$. For each $i \\in \\mathcal I$, let $X_i$ be a matrix of size $n_i \\times m_i$. Suppose $A_{ij} X_j = X_i B_{ij}$ for all $i, j$. We prove versions of Schur's lemma for $\\mathcal A, \\mathcal B$ satisfying coupled irreducibility conditions. We also consider a refinement of Schur's lemma for sets of normal matrices and prove corresponding versions for $\\mathcal A, \\mathcal B$ satisfying coupled normality and coupled irreducibility conditions.<\/jats:p>","DOI":"10.1137\/18m1232462","type":"journal-article","created":{"date-parts":[[2019,9,5]],"date-time":"2019-09-05T11:28:08Z","timestamp":1567682888000},"page":"998-1021","source":"Crossref","is-referenced-by-count":1,"title":["Schur's Lemma for Coupled Reducibility and Coupled Normality"],"prefix":"10.1137","volume":"40","author":[{"given":"Dana","family":"Lahat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Jutten","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helene","family":"Shapiro","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,9,5]]},"reference":[{"key":"atypb1","unstructured":"T. Adal\\i, D. Lahat, and C. Jutten,\n                      Data Fusion: Benefits of Fully Exploiting Diversity\n                      , tutorial presented at\n                      EUSIPCO\n                      , Aug. 2016."},{"key":"atypb2","unstructured":"M. Artin,\n                      Algebra\n                      , 2 ed., Pearson Prentice Hall, NJ, 2011."},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"R. A. Brualdi and H. J. Ryser,\n                      Combinatorial Matrix Theory\n                      , Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1991,https:\/\/doi.org\/10.1017\/CBO9781107325708.","DOI":"10.1017\/CBO9781107325708"},{"key":"atypb4","first-page":"1941","author":"Cardoso J.-F.","year":"1998","journal-title":"WA"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"A. Cichocki and S. Amari,\n                      Adaptive Blind Signal and Image Processing. Learning Algorithms and Applications\n                      , John Wiley & Sons, New York, 2002.","DOI":"10.1002\/0470845899"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1684(94)90029-9"},{"key":"atypb7","unstructured":"P. Comon and C. Jutten,\n                      Handbook of Blind Source Separation: Independent Component Analysis and Applications\n                      , 1st ed., Academic Press, Cambridge, 2010."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/070690729"},{"key":"atypb9","first-page":"370","author":"Gutch H. W.","year":"2010","journal-title":"Heidelberg"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"A. Hyv\u00e4rinen, J. Karhunen, and E. Oja,\n                      Independent Component Analysis\n                      , John Wiley & Sons, New York, 2001.","DOI":"10.1002\/0471221317"},{"key":"atypb11","first-page":"165","author":"Kim T.","year":"2006","journal-title":"Heidelberg"},{"key":"atypb12","unstructured":"D. Lahat,\n                      A Data Fusion Perspective on Source Separation\n                      , talk presented at the Hausdorff School: Low-rank Tensor Techniques in Numerical Analysis and Optimization, Bonn, Germany, Apr. 2016."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2015.2460697"},{"key":"atypb14","first-page":"1875","author":"Lahat D.","year":"2012","journal-title":"Romania"},{"key":"atypb15","first-page":"1417","author":"Lahat D.","year":"2014","journal-title":"Portugal"},{"key":"atypb16","unstructured":"D. Lahat and C. Jutten,\n                      A Generalization to Schur's Lemma with an Application to Joint Independent Subspace Analysis\n                      , Tech. Report, hal-01247899, GIPSA-Lab, Grenoble, France, 2015,https:\/\/hal.archives-ouvertes.fr\/hal-01247899."},{"key":"atypb17","unstructured":"D. Lahat and C. Jutten,\n                      On the uniqueness of coupled matrix block diagonalization in the joint analysis of multiple datasets (talk)\n                      , SIAM Conference on Applied Linear Algebra, Oct. 2015,http:\/\/meetings.siam.org\/sess\/dsp_talk.cfm?p=72077."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2016.2526960"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"D. Lahat and C. Jutten,\n                      Joint independent subspace analysis by coupled block decomposition: Non-identifiable cases\n                      , in Proc. ICASSP, Calgary, Canada, Apr. 2018.","DOI":"10.1109\/ICASSP.2018.8461358"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2018.2880714"},{"key":"atypb21","unstructured":"T. Maehara and K. Murota,\n                      Algorithm for Error-Controlled Simultaneous Block-Diagonalization of Matrices\n                      , Tech. Report METR 2009-53, Department of Mathematical Informatics, Graduate School of Information Science and Technology, the University of Tokyo, Bunkyo-Ku, Tokyo 113-8656, Japan, Dec. 2009."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-010-0006-9"},{"key":"atypb23","unstructured":"I. Schur,\n                      Neue Begr\u00fcndung der Theorie der Gruppencharaktere\n                      , in Sitzungsberichte der K\u00f6niglich-Preussischen Akademie der Wissenschaften zu Berlin, Berlin, Germany, Jan.Jun. 1905, pp. 406-432,https:\/\/archive.org\/details\/sitzungsberichte1905deutsch."},{"key":"atypb24","first-page":"2864","author":"Silva R. F.","year":"2014","journal-title":"France"},{"key":"atypb25","first-page":"67","volume":"99","author":"Sylvester J. J.","year":"1884","journal-title":"Sci."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1232462","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:36:12Z","timestamp":1787326572000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1232462"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1232462"],"URL":"https:\/\/doi.org\/10.1137\/18m1232462","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}