Schur's Lemma for Coupled Reducibility and Coupled Normality
| Autor: | Dana Lahat, Christian Jutten, Helene Shapiro |
|---|---|
| Přispěvatelé: | Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, GIPSA - Vision and Brain Signal Processing (GIPSA-VIBS), Département Images et Signal (GIPSA-DIS), Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Department of Mathematics and Statistics, Swarthmore College, Swarthmore College, European Project: 320684,EC:FP7:ERC,ERC-2012-ADG_20120216,CHESS(2013), European Project: 681839,H2020,ERC-2015-CoG,FACTORY(2016), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
joint independent subspace analysis (JISA)
Mathematics::Number Theory High Energy Physics::Lattice media_common.quotation_subject Schur's lemma coupled reducibility Normal matrix Combinatorics Schur's Lemma FOS: Mathematics normal matrices Normality Mathematics media_common Condensed Matter::Quantum Gases [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] coupled normality Mathematics - Rings and Algebras 15A04 15A21 15A24 15A99 20C99 Indexed family Sylvester equation Rings and Algebras (math.RA) Index set coupled decomposition High Energy Physics::Experiment independent component analysis (ICA) Analysis |
| Zdroj: | SIAM Journal on Matrix Analysis and Applications SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2019, 40 (3), pp.998-1021. ⟨10.1137/18M1232462⟩ SIAM Journal on Matrix Analysis and Applications, 2019, 40 (3), pp.998-1021. ⟨10.1137/18M1232462⟩ |
| ISSN: | 0895-4798 1095-7162 |
| DOI: | 10.1137/18M1232462⟩ |
| Popis: | 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. 35 pages. Second version corrects some typos in the original submission and makes some changes in MSC classification numbers |
| Databáze: | OpenAIRE |
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