{"title":"Multi-granularity co-clustering of multi-type data via symmetric nonnegative matrix factorization","authors":"Dongjin Choi, Haesun Park","doi":"10.1016/j.laa.2025.10.028","DOIUrl":"10.1016/j.laa.2025.10.028","url":null,"abstract":"<div><div>Clustering is a fundamental task in data analysis, essential for discovering patterns and groupings in data. When dealing with multi-type data, where entities of different types are interrelated, clustering becomes more complex and requires specialized methods. Most existing clustering approaches focus on a single type of entity, potentially overlooking the rich interactions between different types. Co-clustering methods address this limitation by simultaneously clustering multiple types of entities, exploiting their interrelationships. However, current co-clustering methods may not fully capture the multi-granularity structures present in many real-world data sets, where clusters exist at varying levels of granularity.</div><div>To address this issue, we propose MG-NMF (Multi-Granularity Nonnegative Matrix Factorization), a method for multi-granularity co-clustering of multi-type data. MG-NMF integrates both intra-type and inter-type relationships through embedding entities of different types into a shared low-dimensional space. By taking as input an integrated symmetric similarity matrix that encodes the relationships among all entity types, MG-NMF simultaneously considers intra-type similarities within each type and inter-type similarities across different types. Furthermore, the framework incorporates a multi-granularity perspective, enabling the discovery of cluster structures at varying levels of granularity, from broader to more refined groupings.</div><div>The proposed method employs a symmetric nonnegative matrix factorization to obtain nonnegative embeddings in a shared space. The nonnegativity constraint ensures interpretability and captures the inherent clustering structure of the data. We present an optimization procedure based on block coordinate descent and provide convergence analysis.</div><div>We evaluate the proposed method on real-world data sets, including a hierarchical data set of scholarly entities. Experimental results indicate that MG-NMF captures hierarchical relationships between clusters at different granularity levels and achieves high-quality clustering performance. MG-NMF offers a unified framework for multi-granularity co-clustering of multi-type data, providing insights into the complex structures of real-world data sets.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"731 ","pages":"Pages 252-276"},"PeriodicalIF":1.1,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145621576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A linear condition for non-very generic discriminantal arrangements","authors":"Simona Settepanella , So Yamagata","doi":"10.1016/j.laa.2025.10.037","DOIUrl":"10.1016/j.laa.2025.10.037","url":null,"abstract":"<div><div>The discriminantal arrangement is the space of configurations of <em>n</em> hyperplanes in generic position in a <em>k</em>-dimensional space. Unlike the case <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, where it coincides with the well-known braid arrangement, the discriminantal arrangement for <span><math><mi>k</mi><mo>></mo><mn>1</mn></math></span> has a combinatorial structure that depends on the choice of the original <em>n</em> hyperplanes. It is known that this combinatorics remains constant on a Zariski-open set <span><math><mi>Z</mi></math></span>, but determining whether a given configuration of <em>n</em> generic hyperplanes belongs to <span><math><mi>Z</mi></math></span> has proved to be a nontrivial problem. Even providing explicit examples of configurations not contained in <span><math><mi>Z</mi></math></span> remains a challenging task. In this paper, building on a recent result by the present authors, we introduce the notion of <em>weak linear independence</em> among sets of vectors, which, when imposed, allows us to construct configurations of hyperplanes not lying in <span><math><mi>Z</mi></math></span>. We also present three explicit examples illustrating this construction.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 603-616"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145517198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral extremal problem for the odd prism","authors":"Xinhui Duan, Lu Lu","doi":"10.1016/j.laa.2025.10.025","DOIUrl":"10.1016/j.laa.2025.10.025","url":null,"abstract":"<div><div>The spectral Turán number <span><math><mtext>spex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> denotes the maximum spectral radius <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of an <em>F</em>-free graph <em>G</em> of order <em>n</em>. This paper determines <span><math><mtext>spex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup><mo>)</mo></mrow></math></span> for sufficiently large <em>n</em>, establishing the unique extremal graph. Here, <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup></math></span> is the odd prism, which is the Cartesian product <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, where the Cartesian product <span><math><mi>G</mi><mo>□</mo><mi>F</mi></math></span> has vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>×</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, and edges between <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> if either <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, or <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 276-296"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Frederik vom Ende , Dariusz Chruściński , Gen Kimura , Paolo Muratore-Ginanneschi
{"title":"Universal bound on the eigenvalues of 2-positive trace-preserving maps","authors":"Frederik vom Ende , Dariusz Chruściński , Gen Kimura , Paolo Muratore-Ginanneschi","doi":"10.1016/j.laa.2025.10.022","DOIUrl":"10.1016/j.laa.2025.10.022","url":null,"abstract":"<div><div>We prove an upper bound on the trace of any 2-positive, trace-preserving map in terms of its smallest eigenvalue. We show that this spectral bound is tight, and that 2-positivity is necessary for this inequality to hold in general. Moreover, we use this to infer a similar bound for generators of one-parameter semigroups of 2-positive trace-preserving maps. With this approach we generalize known results for completely positive trace-preserving dynamics while providing a significantly simpler proof that is entirely algebraic.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 262-275"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juan C. Gutierrez Fernandez , E.O. Quintero Vanegas
{"title":"Corrigendum to “Nilpotent linear spaces and Albert's Problem” [Linear Algebra Appl. 518 (2017) 57–78]","authors":"Juan C. Gutierrez Fernandez , E.O. Quintero Vanegas","doi":"10.1016/j.laa.2025.10.010","DOIUrl":"10.1016/j.laa.2025.10.010","url":null,"abstract":"<div><div>In our article Nilpotent Linear Spaces and Albert's Problem [Linear Algebra Appl. 518 (2017) 57–78], the proof of Theorem 6 was incomplete, as a case was omitted. Here we supply the missing argument. The statement of Theorem 6, and all subsequent results depending on it, remain valid.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 313-317"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform hypertrees with maximum nullity","authors":"Ya-Nan Zheng","doi":"10.1016/j.laa.2025.10.016","DOIUrl":"10.1016/j.laa.2025.10.016","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> be the adjacency tensor of a <em>k</em>-uniform hypergraph <em>H</em>. The nullity of <em>H</em> is the multiplicity of the eigenvalue zero in the spectrum of <em>H</em>, i.e., the algebraic multiplicity of the eigenvalue zero of <span><math><mi>A</mi></math></span>. A connected and acyclic hypergraph is called a hypertree. In this paper, by exploring the relationship between the nullity of <em>k</em>-uniform hypertrees and the nullity of their subhypergraphs, we study the extremal nullity of <em>k</em>-uniform hypertrees. We prove that the <em>k</em>-uniform hyperstar <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> attains uniquely the maximum nullity among all <em>k</em>-uniform hypertrees with <em>m</em> edges.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 138-151"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniel A. Jaume , Cristian Panelo , Maikon M. Toledo , Micaela E. Vega
{"title":"On the core-nilpotent decomposition of unicyclic graphs","authors":"Daniel A. Jaume , Cristian Panelo , Maikon M. Toledo , Micaela E. Vega","doi":"10.1016/j.laa.2025.10.034","DOIUrl":"10.1016/j.laa.2025.10.034","url":null,"abstract":"<div><div>In this work we show, through the null decomposition of unicyclic graphs given by Allem et al. (2020), that the core-nilpotent decomposition of the adjacency matrix of a unicyclic graph, can be obtained directly from the graph itself.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 498-524"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The main reasons for matrices multiplying to zero","authors":"Jakub Koncki , Richárd Rimányi","doi":"10.1016/j.laa.2025.11.001","DOIUrl":"10.1016/j.laa.2025.11.001","url":null,"abstract":"<div><div>We provide an explicit description of the maximal-dimensional components of the variety parametrizing sequences of matrices of prescribed sizes whose product is zero.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 587-602"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The signless Laplacian spectral Turán problems for color-critical graphs","authors":"Jian Zheng , Yongtao Li , Honghai Li","doi":"10.1016/j.laa.2025.10.036","DOIUrl":"10.1016/j.laa.2025.10.036","url":null,"abstract":"<div><div>The well-known Turán theorem states that if <em>G</em> is an <em>n</em>-vertex <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph, then <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo></math></span>, with equality if and only if <em>G</em> is the <em>r</em>-partite Turán graph <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span>. A graph <em>F</em> is called color-critical if it contains an edge whose deletion reduces its chromatic number. Extending the Turán theorem, Simonovits (1968) proved that for any color-critical graph <em>F</em> with <span><math><mi>χ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn></math></span> and sufficiently large <em>n</em>, the Turán graph <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span> is the unique graph that attains the maximum number of edges over all <em>n</em>-vertex <em>F</em>-free graphs. Subsequently, Nikiforov (2009) <span><span>[40]</span></span> proved a spectral version of Simonovits' theorem in terms of the adjacency spectral radius. In this paper, we show an extension of Simonovits' theorem for the signless Laplacian spectral radius. We prove that for any color-critical graph <em>F</em> with <span><math><mi>χ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>4</mn></math></span>, if <em>n</em> is sufficiently large and <em>G</em> is an <em>F</em>-free graph on <em>n</em> vertices, then <span><math><mi>q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>q</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo></math></span>, with equality if and only if <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span>. Our approach is to establish a signless Laplacian spectral version of the criterion of Keevash, Lenz and Mubayi (2014) <span><span>[26]</span></span>. Consequently, we determine the signless Laplacian spectral extremal graphs for generalized books and even wheels. As an application, our result gives an upper bound on the degree power of an <em>F</em>-free graph. We show that if <em>n</em> is sufficiently large and <em>G</em> is an <em>F</em>-free graph on <em>n</em> vertices with <em>m</em> edges, then <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo><mi>m</mi><mi>n</mi></math></span>, with equality if and only if <em>G</em> is a regular T","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 546-565"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-linear maps preserving ascent or descent of product of operators","authors":"Rabi Marzouki, Khalid Souilah","doi":"10.1016/j.laa.2025.10.026","DOIUrl":"10.1016/j.laa.2025.10.026","url":null,"abstract":"<div><div>In this article, we provide a complete description of all maps on the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, that leave invariant the ascent, or descent, under the product of two operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 344-357"},"PeriodicalIF":1.1,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}