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}
Pub Date : 2025-10-29DOI: 10.1016/j.laa.2025.10.032
Sarah Chehade , Andrea Delgado , Shuzhou Wang , Zhenhua Wang
In quantum computing, Trotter estimates are critical for enabling efficient simulation of quantum systems and quantum dynamics, help implement complex quantum algorithms, and provide a systematic way to control approximate errors. In this paper, we extend the analysis of Trotter-Suzuki approximations, including third and higher orders, to Jordan-Banach algebras. We solve an open problem in our earlier paper on the existence of second-order Trotter formula error estimation in Jordan-Banach algebras. To illustrate our work, we apply our formula to simulate Trotter-factorized spins, and show improvements in the approximations. Our approach demonstrates the adaptability of Trotter product formulas and estimates to non-associative settings, which offers new insights into the applications of Jordan algebra theory to operator dynamics.
{"title":"Error estimates and higher order Trotter product formulas in Jordan-Banach algebras","authors":"Sarah Chehade , Andrea Delgado , Shuzhou Wang , Zhenhua Wang","doi":"10.1016/j.laa.2025.10.032","DOIUrl":"10.1016/j.laa.2025.10.032","url":null,"abstract":"<div><div>In quantum computing, Trotter estimates are critical for enabling efficient simulation of quantum systems and quantum dynamics, help implement complex quantum algorithms, and provide a systematic way to control approximate errors. In this paper, we extend the analysis of Trotter-Suzuki approximations, including third and higher orders, to Jordan-Banach algebras. We solve an open problem in our earlier paper on the existence of second-order Trotter formula error estimation in Jordan-Banach algebras. To illustrate our work, we apply our formula to simulate Trotter-factorized spins, and show improvements in the approximations. Our approach demonstrates the adaptability of Trotter product formulas and estimates to non-associative settings, which offers new insights into the applications of Jordan algebra theory to operator dynamics.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"730 ","pages":"Pages 430-449"},"PeriodicalIF":1.1,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418536","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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