{"title":"Multilayer crisscross error and erasure correction","authors":"Umberto Martínez-Peñas","doi":"10.1016/j.laa.2025.09.014","DOIUrl":"10.1016/j.laa.2025.09.014","url":null,"abstract":"<div><div>In this work, multilayer crisscross errors and erasures are considered, which affect entire rows and columns in the matrices of a list of matrices. To measure such errors and erasures, the multi-cover metric is introduced. Several bounds are derived, including a Singleton bound, and maximum multi-cover distance (MMCD) codes are defined as those attaining it. Duality, puncturing and shortening of linear MMCD codes are studied. It is shown that the dual of a linear MMCD code is not necessarily MMCD, and those satisfying this duality condition are defined as dually MMCD codes. Finally, some constructions of codes in the multi-cover metric are given, including dually MMCD codes, together with efficient decoding algorithms for them.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 349-375"},"PeriodicalIF":1.1,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118175","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}
Sooyeong Kim , David Kribs , Edison Lozano , Rajesh Pereira , Sarah Plosker
{"title":"Quasiorthogonality of commutative algebras, complex Hadamard matrices, and mutually unbiased measurements","authors":"Sooyeong Kim , David Kribs , Edison Lozano , Rajesh Pereira , Sarah Plosker","doi":"10.1016/j.laa.2025.09.010","DOIUrl":"10.1016/j.laa.2025.09.010","url":null,"abstract":"<div><div>We deepen the theory of quasiorthogonal and approximately quasiorthogonal operator algebras through an analysis of the commutative algebra case. We give a new approach to calculate the measure of orthogonality between two such subalgebras of matrices, based on a matrix-theoretic notion we introduce that has a connection to complex Hadamard matrices. We also show how this new tool can yield significant information on the general non-commutative case. We finish by considering quasiorthogonality for the important subclass of commutative algebras that arise from mutually unbiased bases (MUBs) and mutually unbiased measurements (MUMs) in quantum information theory. We present a number of examples throughout the work, including a subclass that arises from group algebras and Latin squares.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 383-408"},"PeriodicalIF":1.1,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118083","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":"Generalized spectral characterization of signed bipartite graphs","authors":"Songlin Guo , Wei Wang , Lele Li","doi":"10.1016/j.laa.2025.09.008","DOIUrl":"10.1016/j.laa.2025.09.008","url":null,"abstract":"<div><div>Let Σ be an <em>n</em>-vertex controllable or almost controllable signed bipartite graph, and let <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>Σ</mi></mrow></msub></math></span> denote the discriminant of its characteristic polynomial <span><math><mi>χ</mi><mo>(</mo><mi>Σ</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>. We prove that if (i) the integer <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></msup><msqrt><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>Σ</mi></mrow></msub></mrow></msqrt></math></span> is squarefree, and (ii) the constant term (even <em>n</em>) or linear coefficient (odd <em>n</em>) of <span><math><mi>χ</mi><mo>(</mo><mi>Σ</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span> is ±1, then Σ is determined by its generalized spectrum. This result extends a recent theorem of Ji et al. (2025) <span><span>[6]</span></span>, which established a similar criterion for signed trees with irreducible characteristic polynomials.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 331-348"},"PeriodicalIF":1.1,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096353","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}
Limeng Lin , Luiz Emilio Allem , Vilmar Trevisan , Wei Wang , Hao Zhang
{"title":"A family of graphs that are DGS but not DS","authors":"Limeng Lin , Luiz Emilio Allem , Vilmar Trevisan , Wei Wang , Hao Zhang","doi":"10.1016/j.laa.2025.09.009","DOIUrl":"10.1016/j.laa.2025.09.009","url":null,"abstract":"<div><div>The spectral characterization of graphs is a central theme in spectral graph theory. A graph <em>G</em> is <em>determined by its spectrum</em> (DS) if every graph cospectral with <em>G</em> is also isomorphic to <em>G</em>. The definition is extended to the generalized spectrum, where a graph <em>G</em> is <em>determined by its generalized spectrum</em> (DGS) if any graph <em>H</em> that is cospectral with <em>G</em> and whose complement is cospectral with <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> must be isomorphic to <em>G</em>. While it is clear that all DS graphs are also DGS, the reverse is not always true. This leads to a natural, unanswered question: Which graphs are DGS but not DS? Previous research has focused on identifying graphs that are either DS or DGS, but, to our knowledge, research on this specific problem has not attracted much attention. This paper addresses the problem by introducing an infinite family of graphs that are DGS but not DS.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 283-294"},"PeriodicalIF":1.1,"publicationDate":"2025-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096342","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":"Eigenvalues and signature of quadratic forms associated with finite topological spaces","authors":"Pedro J. Chocano","doi":"10.1016/j.laa.2025.09.007","DOIUrl":"10.1016/j.laa.2025.09.007","url":null,"abstract":"<div><div>Given any finite topological space <em>X</em> and a field <span><math><mi>K</mi></math></span>, we associate a quadratic space <span><math><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, consisting of a vector space <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> over <span><math><mi>K</mi></math></span> and a quadratic form <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>:</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>×</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>→</mo><mi>K</mi></math></span>, to <em>X</em>. The eigenvalues and signature of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> are topological invariants of <em>X</em>. We study their relations with <em>X</em>. From this, we obtain restrictions to check whether a finite topological space can be embedded into another one. Additionally, we compute these invariants for minimal finite models of spheres and other families of finite spaces.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 263-282"},"PeriodicalIF":1.1,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096341","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":"On the incidence matrix of a graph with matrix weights","authors":"Madhab Mondal, Sukanta Pati, Bhaba Kumar Sarma","doi":"10.1016/j.laa.2025.09.003","DOIUrl":"10.1016/j.laa.2025.09.003","url":null,"abstract":"<div><div>Let <em>G</em> be a simple, oriented, edge weighted graph with <em>n</em> vertices and <em>m</em> edges, where weights are matrices in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> (the set of all square matrices of order <em>s</em>). It is well-known that if <em>G</em> is connected and weights are nonzero scalars, then the rank of the <em>vertex-edge incidence matrix</em> <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We observe that if the weights are rank <em>k</em> matrices in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>, then <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>≤</mo><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo></mo><mo>{</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>s</mi><mo>,</mo><mi>m</mi><mi>k</mi><mo>}</mo></math></span>. In particular, when <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span> (i.e., weights as rank one matrices) and <span><math><mi>s</mi><mo>≥</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span>, then <span><math><mi>n</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>m</mi></math></span>. We show that for large values of <em>s</em>, there exist assignments of rank one weights from <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> such that all integer values between <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> and <em>m</em> can be attained as the rank of <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Further, we study the smallest possible values of <em>s</em> for which these ranks can be attained. Surprisingly, the smallest value of <em>s</em> for which <em>m</em> can be achieved as <span><math><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> need not be <span><math><mo>⌈</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo></math></span>. Even more interestingly, it turns out that the minimum value of <em>s</em> is the arboricity of the graph, i.e., the least number of colors needed to color the edges of <em>G</em> so that no cycle is monochromatic. As an extension, we supply an expression of the minimum values of <em>s</em> for which the intermediate values for the ranks of <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can be achieved. For a graph <em>G</em>, <span><math><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can give more information on <em>G</em> if we consider matrix weights. We show that a connected graph <em>G</em> on <em>n</em> vertices is a tree if and only if for every assignment of rank one weights from <sp","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 295-319"},"PeriodicalIF":1.1,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096351","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":"(Positive) quadratic determinantal representations of quartic curves and the Robinson polynomial","authors":"Clemens Brüser, Mario Kummer","doi":"10.1016/j.laa.2025.09.006","DOIUrl":"10.1016/j.laa.2025.09.006","url":null,"abstract":"<div><div>We prove that every real nonnegative ternary quartic whose complex zero set is smooth can be represented as the determinant of a symmetric matrix with quadratic entries which is everywhere positive semidefinite. We show that the corresponding statement fails for the Robinson polynomial, answering a question by Buckley and Šivic.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 232-262"},"PeriodicalIF":1.1,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060416","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":"Almost commuting self-adjoint operators and iterated commutator estimates","authors":"Jakob Geisler","doi":"10.1016/j.laa.2025.09.004","DOIUrl":"10.1016/j.laa.2025.09.004","url":null,"abstract":"<div><div>Given two almost commuting self-adjoint operators, a new method for finding exactly commuting operators is presented. For this, a differential equation for self-adjoint Hilbert-Schmidt operators is introduced. Quantitative results are proven that the exactly commuting operators are close to the old ones in the Hilbert-Schmidt norm. The proof relies on a novel estimate in which the norm of the commutator is bounded from above by the norm of the iterated commutators times a constant. This inequality is proven in finite dimensions and lower bounds for the optimal constants are given.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"727 ","pages":"Pages 388-411"},"PeriodicalIF":1.1,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018511","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":"Hodge operators and groups of isometries of diagonalizable symmetric bilinear forms in characteristic two","authors":"Linus Kramer , Markus J. Stroppel","doi":"10.1016/j.laa.2025.09.002","DOIUrl":"10.1016/j.laa.2025.09.002","url":null,"abstract":"<div><div>We study groups of isometries of non-alternating symmetric bilinear forms on vector spaces of characteristic two, and actions of these groups on exterior powers of the space, viewed as modules over algebras generated by Hodge operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 211-231"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145045933","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":"On Araki-type trace inequalities","authors":"Po-Chieh Liu , Hao-Chung Cheng","doi":"10.1016/j.laa.2025.08.023","DOIUrl":"10.1016/j.laa.2025.08.023","url":null,"abstract":"<div><div>In this paper, we prove a trace inequality <span><math><mi>Tr</mi><mspace></mspace><mrow><mo>[</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></mrow><mo>≤</mo><mi>Tr</mi><mo>[</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>B</mi><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></math></span> for any positive and monotonically increasing function <em>f</em>, <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, and positive semi-definite matrices <em>A</em> and <em>B</em>. On the other hand, if <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and the map <span><math><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>s</mi></mrow></msup><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is positive and decreasing, then <span><math><mi>Tr</mi><mo>[</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mi>B</mi><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo><mo>≤</mo><mi>Tr</mi><mspace></mspace><mrow><mo>[</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>]</mo></mrow></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 320-330"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096352","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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