Pub Date : 2025-11-07DOI: 10.1016/j.laa.2025.10.028
Dongjin Choi, Haesun Park
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.
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.
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.
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.
聚类是数据分析中的一项基本任务，对于发现数据中的模式和分组至关重要。当处理多类型数据时，不同类型的实体是相互关联的，聚类变得更加复杂，需要专门的方法。大多数现有的聚类方法关注于单一类型的实体，可能忽略了不同类型之间的丰富交互。协同聚类方法通过同时聚类多种类型的实体，利用它们的相互关系来解决这一限制。然而，当前的共聚类方法可能无法完全捕获许多真实数据集中存在的多粒度结构，其中集群以不同的粒度级别存在。为了解决这个问题，我们提出了一种多类型数据的多粒度共聚类方法MG-NMF （Multi-Granularity non - negative Matrix Factorization）。MG-NMF通过将不同类型的实体嵌入到共享的低维空间中，整合了类型内关系和类型间关系。MG-NMF以编码所有实体类型之间关系的集成对称相似度矩阵为输入，同时考虑每种类型内的类型相似度和不同类型间的类型相似度。此外，该框架还包含了一个多粒度透视图，允许在不同粒度级别上发现集群结构，从更广泛的到更精细的分组。该方法采用对称非负矩阵分解来获得共享空间中的非负嵌入。非负性约束确保了数据的可解释性，并捕获了数据固有的聚类结构。提出了一种基于分块坐标下降的优化方法，并给出了收敛性分析。我们在真实世界的数据集上评估了所提出的方法，包括学术实体的分层数据集。实验结果表明，MG-NMF捕获了不同粒度级别聚类之间的层次关系，获得了高质量的聚类性能。MG-NMF为多类型数据的多粒度共聚类提供了一个统一的框架，提供了对现实世界数据集复杂结构的见解。
{"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":"2025-11-07","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}
Pub Date : 2025-11-06DOI: 10.1016/j.laa.2025.10.037
Simona Settepanella , So Yamagata
The discriminantal arrangement is the space of configurations of n hyperplanes in generic position in a k-dimensional space. Unlike the case , where it coincides with the well-known braid arrangement, the discriminantal arrangement for has a combinatorial structure that depends on the choice of the original n hyperplanes. It is known that this combinatorics remains constant on a Zariski-open set , but determining whether a given configuration of n generic hyperplanes belongs to has proved to be a nontrivial problem. Even providing explicit examples of configurations not contained in remains a challenging task. In this paper, building on a recent result by the present authors, we introduce the notion of weak linear independence among sets of vectors, which, when imposed, allows us to construct configurations of hyperplanes not lying in . We also present three explicit examples illustrating this construction.
{"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":"2025-11-06","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}
Pub Date : 2025-11-06DOI: 10.1016/j.laa.2025.11.001
Jakub Koncki , Richárd Rimányi
We provide an explicit description of the maximal-dimensional components of the variety parametrizing sequences of matrices of prescribed sizes whose product is zero.
我们给出了乘积为零的规定大小矩阵的各种参数化序列的最大维分量的显式描述。
{"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":"2025-11-06","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}
Pub Date : 2025-11-05DOI: 10.1016/j.laa.2025.10.036
Jian Zheng , Yongtao Li , Honghai Li
<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
{"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":"2025-11-05","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}
Pub Date : 2025-10-31DOI: 10.1016/j.laa.2025.10.029
Hiroshi Hirai
<div><div>Given a tuple of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, the linear symbolic matrix <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is nonsingular in the noncommutative sense if and only if the completely positive operators <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>X</mi><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>†</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>†</mi></mrow></msubsup><mi>X</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> can be scaled to be doubly stochastic: For every <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> there are <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>‖</mo><msub><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo><mo><</mo><mi>ϵ</mi></math></span>, <span><math><mo>‖</mo><msubsup><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo><mo><</mo><mi>ϵ</mi></math></span>. In this paper, we show a refinement: The noncommutative corank of <em>A</em> is equal to one-half of the minimum residual <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><msubsup><mrow><mi>T</mi></mrow><
{"title":"A scaling characterization of nc-rank via unbounded gradient flow","authors":"Hiroshi Hirai","doi":"10.1016/j.laa.2025.10.029","DOIUrl":"10.1016/j.laa.2025.10.029","url":null,"abstract":"<div><div>Given a tuple of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> complex matrices <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, the linear symbolic matrix <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is nonsingular in the noncommutative sense if and only if the completely positive operators <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>X</mi><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>†</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>†</mi></mrow></msubsup><mi>X</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> can be scaled to be doubly stochastic: For every <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> there are <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> such that <span><math><mo>‖</mo><msub><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo><mo><</mo><mi>ϵ</mi></math></span>, <span><math><mo>‖</mo><msubsup><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo><mo><</mo><mi>ϵ</mi></math></span>. In this paper, we show a refinement: The noncommutative corank of <em>A</em> is equal to one-half of the minimum residual <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>g</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>A</mi><mi>h</mi></mrow></msub><mo>(</mo><mi>I</mi><mo>)</mo><mo>−</mo><mi>I</mi><mo>‖</mo></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><msubsup><mrow><mi>T</mi></mrow><","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 525-545"},"PeriodicalIF":1.1,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467945","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}
Pub Date : 2025-10-30DOI: 10.1016/j.laa.2025.10.034
Daniel A. Jaume , Cristian Panelo , Maikon M. Toledo , Micaela E. Vega
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.
{"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":"2025-10-30","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}
Pub Date : 2025-10-30DOI: 10.1016/j.laa.2025.10.033
Vesa Kaarnioja , André-Alexander Zepernick
Let denote the set of all nonsingular lower triangular -matrices. Hong and Loewy (2004) introduced the number sequence There have been a number of attempts in the literature to obtain bounds on the numbers by Mattila (2015), Altınışık et al. (2016), Kaarnioja (2021), Loewy (2021), and Altınışık (2021). In this paper, improved upper and lower bounds are derived for the numbers . By considering the characteristic polynomial corresponding to the matrix satisfying , it is shown that the second largest eigenvalue of is bounded from above by leading to an improved upper bound on . On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds.
设Kn表示所有非奇异n×n下三角(0,1)-矩阵的集合。Hong and Loewy（2004）引入了数sequencecn=min (λ|λ) λ是xxt的特征值，X∈Kn},n∈Z+。文献中已经有许多尝试通过Mattila (2015)， Altınışık等人（2016），Kaarnioja (2021), Loewy（2021）和Altınışık（2021）来获得数字cn的界限。本文导出了数cn的改进上界和下界。通过考虑满足cn=‖Zn‖2−1的矩阵Zn对应的特征多项式，证明了Zn的第二大特征值上界为45，从而得到cn的改进上界。另一方面，将Samuelson不等式应用于Zn的特征多项式的根，得到了改进的下界。数值实验证明了新边界的有效性。
{"title":"New upper and lower bounds on the smallest singular values of nonsingular lower triangular (0,1)-matrices","authors":"Vesa Kaarnioja , André-Alexander Zepernick","doi":"10.1016/j.laa.2025.10.033","DOIUrl":"10.1016/j.laa.2025.10.033","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of all nonsingular <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> lower triangular <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-matrices. Hong and Loewy (2004) introduced the number sequence<span><span><span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mi>λ</mi><mo>|</mo><mi>λ</mi><mspace></mspace><mtext>is an eigenvalue of</mtext><mspace></mspace><mi>X</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>X</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>n</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>.</mo></math></span></span></span> There have been a number of attempts in the literature to obtain bounds on the numbers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by Mattila (2015), Altınışık et al. (2016), Kaarnioja (2021), Loewy (2021), and Altınışık (2021). In this paper, improved upper and lower bounds are derived for the numbers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By considering the characteristic polynomial corresponding to the matrix <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfying <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>‖</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></math></span>, it is shown that the second largest eigenvalue of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is bounded from above by <span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> leading to an improved upper bound on <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 483-497"},"PeriodicalIF":1.1,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418521","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}
Pub Date : 2025-10-30DOI: 10.1016/j.laa.2025.10.030
Anzila Laikhuram , Jephian C.-H. Lin
A discrete Schrödinger operator of a graph G is a real symmetric matrix whose -entry, , is negative if is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete Schrödinger operators have been used to study vibration theory and the Colin de Verdière parameter. The inverse eigenvalue problem for discrete Schrödinger operators of a graph aims to characterize the possible spectra among discrete Schrödinger operators of a graph. Compared to the inverse eigenvalue problem of a graph, the answers turn out to be more limited, and several restrictions based on graph structure are given. Using the strong properties, analogous versions of the supergraph lemma, the liberation lemma, and the bifurcation lemma are established. Using these results, the inverse eigenvalue problem for discrete Schrödinger operators is resolved for each graph with at most 5 vertices.
图G的离散Schrödinger算子是一个实对称矩阵，其i，j项，i≠j，当{i，j}是边时为负，当{i，j}不是边时为零，而对角线项可以是任何实数。离散Schrödinger算符已被用于研究振动理论和Colin de verdi参数。图的离散Schrödinger算子的特征值反问题旨在描述图的离散Schrödinger算子之间可能的谱。与图的特征值反问题相比，该问题的答案更有局限性，并给出了基于图结构的若干限制条件。利用这些强性质，建立了超图引理、解放引理和分岔引理的类似形式。利用这些结果，离散Schrödinger算子的反特征值问题解决了每个最多有5个顶点的图。
{"title":"Inverse eigenvalue problem for discrete Schrödinger operators of a graph","authors":"Anzila Laikhuram , Jephian C.-H. Lin","doi":"10.1016/j.laa.2025.10.030","DOIUrl":"10.1016/j.laa.2025.10.030","url":null,"abstract":"<div><div>A discrete Schrödinger operator of a graph <em>G</em> is a real symmetric matrix whose <span><math><mi>i</mi><mo>,</mo><mi>j</mi></math></span>-entry, <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>, is negative if <span><math><mo>{</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>}</mo></math></span> is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete Schrödinger operators have been used to study vibration theory and the Colin de Verdière parameter. The inverse eigenvalue problem for discrete Schrödinger operators of a graph aims to characterize the possible spectra among discrete Schrödinger operators of a graph. Compared to the inverse eigenvalue problem of a graph, the answers turn out to be more limited, and several restrictions based on graph structure are given. Using the strong properties, analogous versions of the supergraph lemma, the liberation lemma, and the bifurcation lemma are established. Using these results, the inverse eigenvalue problem for discrete Schrödinger operators is resolved for each graph with at most 5 vertices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 566-586"},"PeriodicalIF":1.1,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467946","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}
Pub Date : 2025-10-29DOI: 10.1016/j.laa.2025.10.031
Eman Aldabbas , Mohammad Sababheh
Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix A and any , the matrix is accretive-dissipative, and that where is the numerical radius. This inequality complements the well-known power inequality , valid for any square matrix and positive integer power k. As an application, we prove that if A is accretive, then the above fractional inequality holds if . Other consequences will be given too.
{"title":"The numerical radius of fractional powers of matrices","authors":"Eman Aldabbas , Mohammad Sababheh","doi":"10.1016/j.laa.2025.10.031","DOIUrl":"10.1016/j.laa.2025.10.031","url":null,"abstract":"<div><div>Using integral representations of the fractional power of matrices, and the geometric intuition of sectorial matrices, we show that for any accretive-dissipative matrix <em>A</em> and any <span><math><mi>t</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, the matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span> is accretive-dissipative, and that<span><span><span><math><mi>ω</mi><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>ω</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> is the numerical radius. This inequality complements the well-known power inequality <span><math><mi>ω</mi><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo><mo>≤</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, valid for any square matrix and positive integer power <em>k</em>. As an application, we prove that if <em>A</em> is accretive, then the above fractional inequality holds if <span><math><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≤</mo><mi>t</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Other consequences will be given too.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 450-460"},"PeriodicalIF":1.1,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418518","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}
Pub Date : 2025-10-29DOI: 10.1016/j.laa.2025.10.027
Cairong Chen , Xuehua Li , Ren-Cang Li
An underdetermined generalized absolute value equation (GAVE) may have no solution, one solution, finitely many or infinitely many solutions. This paper is concerned with sufficient conditions that guarantee the existence of solutions to an underdetermined GAVE. Particularly, sufficient conditions are established for an underdetermined GAVE to have infinitely many solutions with no zero entry that possess a particular or any given sign pattern. Some existing results for square GAVE are also extended. It is noted that some of the proofs are constructive and lead to iterative schemes that can solve the underdetermined GAVE in question.
{"title":"Solutions for underdetermined generalized absolute value equations","authors":"Cairong Chen , Xuehua Li , Ren-Cang Li","doi":"10.1016/j.laa.2025.10.027","DOIUrl":"10.1016/j.laa.2025.10.027","url":null,"abstract":"<div><div>An underdetermined generalized absolute value equation (GAVE) may have no solution, one solution, finitely many or infinitely many solutions. This paper is concerned with sufficient conditions that guarantee the existence of solutions to an underdetermined GAVE. Particularly, sufficient conditions are established for an underdetermined GAVE to have infinitely many solutions with no zero entry that possess a particular or any given sign pattern. Some existing results for square GAVE are also extended. It is noted that some of the proofs are constructive and lead to iterative schemes that can solve the underdetermined GAVE in question.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 461-482"},"PeriodicalIF":1.1,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418517","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}
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