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Title: Additive decomposition of matrices under rank conditions and zero pattern constraints (English)
Author: Bart, Harm
Author: Ehrhardt, Torsten
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 72
Issue: 3
Year: 2022
Pages: 825-854
Summary lang: English
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Category: math
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Summary: This paper deals with additive decompositions $A=A_1+\cdots +A_p$ of a given matrix $A$, where the ranks of the summands $A_1,\ldots , A_p$ are prescribed and meet certain zero pattern requirements. The latter are formulated in terms of directed bipartite graphs. (English)
Keyword: additive decomposition
Keyword: rank constraint
Keyword: zero pattern constraint
Keyword: directed bipartite graph
Keyword: $ß{L}$-free directed bipartite graph
Keyword: permutation $ß{L}$-free directed bipartite graph
Keyword: Bell number
Keyword: Stirling partition number
MSC: 05C20
MSC: 05C50
MSC: 15A03
MSC: 15A21
idZBL: Zbl 07584105
idMR: MR4467945
DOI: 10.21136/CMJ.2022.0185-21
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Date available: 2022-08-22T08:24:45Z
Last updated: 2024-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/150620
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Reference: [1] Bart, H., Ehrhardt, T., Silbermann, B.: Rank decomposition in zero pattern matrix algebras.Czech. Math. J. 66 (2016), 987-1005. Zbl 1413.15010, MR 3556880, 10.1007/s10587-016-0305-7
Reference: [2] Bart, H., Ehrhardt, T., Silbermann, B.: Echelon type canonical forms in upper triangular matrix algebras.Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics Operator Theory: Advances and Applications 259. Birkhäuser, Basel (2017), 79-124. Zbl 1365.15014, MR 3644514, 10.1007/978-3-319-49182-0_8
Reference: [3] Bart, H., Ehrhardt, T., Silbermann, B.: $ß{L}$-free directed bipartite graphs and echelon-type canonical forms.Operator Theory, Analysis and the State Space Approach Operator Theory: Advances and Applications 271. Birkhäuser, Cham (2018), 75-117. Zbl 1427.15016, MR 3889652, 10.1007/978-3-030-04269-1_3
Reference: [4] Bart, H., Ehrhardt, T., Silbermann, B.: Rank decomposition under zero pattern constraints and $ß{L}$-free directed graphs.Linear Algebra Appl. 621 (2021), 135-180. Zbl 1464.15002, MR 4231570, 10.1016/j.laa.2021.03.010
Reference: [5] Bart, H., Wagelmans, A. P. M.: An integer programming problem and rank decomposition of block upper triangular matrices.Linear Algebra Appl. 305 (2000), 107-129. Zbl 0951.15013, MR 1733797, 10.1016/S0024-3795(99)00219-0
Reference: [6] Birkhoff, G.: Lattice Theory.American Mathematical Society Colloquium Publications 25. AMS, Providence (1967). Zbl 0153.02501, MR 0227053, 10.1090/coll/025
Reference: [7] Charalambides, C. A.: Enumerative Combinatorics.CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton (2002). Zbl 1001.05001, MR 1937238, 10.1201/9781315273112
Reference: [8] Habib, M., Jegou, R.: $N$-free posets as generalizations of series-parallel posets.Discrete Appl. Math. 12 (1985), 279-291. Zbl 0635.06002, MR 0813975, 10.1016/0166-218X(85)90030-7
Reference: [9] Riordan, J.: Combinatorial Identities.John Wiley & Sons, New York (1968). Zbl 0194.00502, MR 0231725
Reference: [10] Stanley, R. P.: Enumerative Combinatorics. Vol. 1.Cambridge Studies in Advanced Mathematics 49. Cambridge University Press, Cambridge (1997). Zbl 0889.05001, MR 1442260, 10.1017/CBO9780511805967
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