Linear Algebra and its Applications最新文献

{"title":"When is a subspace of ℓ∞N isometrically isomorphic to ℓ∞n?","authors":"Beata Derȩgowska ,&nbsp;Simon Foucart ,&nbsp;Barbara Lewandowska","doi":"10.1016/j.laa.2025.12.009","DOIUrl":"10.1016/j.laa.2025.12.009","url":null,"abstract":"<div><div>It is shown in this note that one can decide whether an <em>n</em>-dimensional subspace of <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> is isometrically isomorphic to <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> by testing a finite number of determinental inequalities. As a byproduct, an elementary proof is provided for the fact that an <em>n</em>-dimensional subspace of <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span> with projection constant equal to one must be isometrically isomorphic to <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 171-177"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788849","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":"Matrix best approximation in the spectral norm","authors":"Vance Faber ,&nbsp;Jörg Liesen ,&nbsp;Petr Tichý","doi":"10.1016/j.laa.2025.12.007","DOIUrl":"10.1016/j.laa.2025.12.007","url":null,"abstract":"<div><div>We derive, similar to Lau and Riha in <span><span>[22]</span></span>, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the <em>k</em>-dimensional field of <em>k</em> matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal for <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> block diagonal matrices containing two identical diagonal blocks. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 178-204"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The L'vov-Kaplansky conjecture for polynomials of degree three","authors":"Daniel Vitas","doi":"10.1016/j.laa.2025.12.016","DOIUrl":"10.1016/j.laa.2025.12.016","url":null,"abstract":"<div><div>The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial <em>f</em> in the matrix algebra <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo></math></span> is a vector space for every <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. We prove this conjecture for the case where <em>f</em> has degree 3 and <em>K</em> is an algebraically closed field of characteristic 0.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 205-232"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A sharp spectral extremal result for general non-bipartite graphs","authors":"John Byrne","doi":"10.1016/j.laa.2025.12.010","DOIUrl":"10.1016/j.laa.2025.12.010","url":null,"abstract":"<div><div>For a graph family <span><math><mi>F</mi></math></span>, let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> denote the maximum number of edges and maximum spectral radius of an <em>n</em>-vertex <span><math><mi>F</mi></math></span>-free graph, respectively, and let <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</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><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> then <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> for <em>n</em> large enough. Fang, Tait, and Zhai extended this result by showing if <span><math><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≤</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>&lt;</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><mo>+</mo><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mi>r</mi><mo>⌋</mo></math></span> then <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> for <em>n</em> large enough, and asked for the maximum constant <span><math><mi>c</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> such that <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</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><mo>+</mo><mo>(</mo><mi>c</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>−</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> guarantees such containment. In this paper we determine <span><math><mi>c</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> exactly for all <span><math><mi>r</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 75-115"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788852","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":"Curves and spectrum localization for real matrices","authors":"Aikaterini Aretaki ,&nbsp;Maria Adam ,&nbsp;Michael Tsatsomeros","doi":"10.1016/j.laa.2025.12.013","DOIUrl":"10.1016/j.laa.2025.12.013","url":null,"abstract":"<div><div>It is well known that the eigenvalues of a complex matrix <em>A</em> are located to the left of the vertical line passing through the largest eigenvalue of its Hermitian part, <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. Adam and Tsatsomeros in <span><span>[1]</span></span> defined a cubic algebraic curve, known as the <em>shell</em> <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of <em>A</em>, using the two largest eigenvalues of <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. This curve localizes the spectrum further and lies to the left of the aforementioned vertical line. Later, Bergqvist in <span><span>[5]</span></span> extended the methodology employed in <span><span>[1]</span></span> to define a new curve, <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, in terms of the three largest eigenvalues of <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. This article delves into the geometry of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for a real matrix <em>A</em> to address some open questions raised in <span><span>[5]</span></span>. In particular, specific conditions are established to characterize the configurations of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> in certain cases. Additionally, the number of eigenvalues of <em>A</em> surrounded by a bounded branch of the curve is examined. Examples are used to validate our findings and demonstrate the quality of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> as a finer spectrum localization area when compared to <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 116-154"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145788853","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":"Fixed points of personalized PageRank centrality: From irreducible to reducible networks","authors":"David Aleja ,&nbsp;Julio Flores ,&nbsp;Eva Primo ,&nbsp;Daniel Rodríguez ,&nbsp;Miguel Romance","doi":"10.1016/j.laa.2025.12.014","DOIUrl":"10.1016/j.laa.2025.12.014","url":null,"abstract":"<div><div>In this paper we analyze PageRank of a complex network as a function of its personalization vector. By using this approach, a complete characterization of the existence and uniqueness of fixed points of the PageRank of a graph is given in terms of the number and nature of its strongly connected components. The method presented essentially follows the classic <em>Power's Method</em> by means of a <em>feedback-PageRank</em> that allows to precisely compute the fixed points, in terms of the (left-hand) Perron vector of each strongly connected component.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"733 ","pages":"Pages 233-272"},"PeriodicalIF":1.1,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880818","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":"Colorful positive bases decomposition and Helly-type results for cones","authors":"Grigory Ivanov","doi":"10.1016/j.laa.2025.11.023","DOIUrl":"10.1016/j.laa.2025.11.023","url":null,"abstract":"<div><div>We prove the following colorful Helly-type result: Fix <span><math><mi>k</mi><mo>∈</mo><mo>[</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>. Assume <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi><mo>+</mo><mo>(</mo><mi>d</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msub></math></span> are finite sets (colors) of nonzero vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. If for every rainbow sub-selection <em>R</em> from these sets of size at most <span><math><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>(</mo><mi>d</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span>, the system <span><math><mrow><mo>〈</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow><mo>≤</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>a</mi><mo>∈</mo><mi>R</mi></math></span> has at least <em>k</em> linearly independent solutions, then at least one of the systems <span><math><mrow><mo>〈</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>〉</mo></mrow><mo>≤</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>a</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>d</mi><mo>+</mo><mo>(</mo><mi>d</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>]</mo></math></span> has at least <em>k</em> linearly independent solutions.</div><div>A <em>rainbow sub-selection</em> from several sets refers to choosing at most one element from each set (color).</div><div>The Helly number <span><math><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>(</mo><mi>d</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span> and the number of colors <span><math><mi>d</mi><mo>+</mo><mo>(</mo><mi>d</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span> are optimal.</div><div>Our key observation is a certain colorful Carathéodory-type result for positive bases.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"732 ","pages":"Pages 108-125"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on almost cospectrality of components of NEPS of bipartite graphs","authors":"Ivan Stanković","doi":"10.1016/j.laa.2025.11.019","DOIUrl":"10.1016/j.laa.2025.11.019","url":null,"abstract":"<div><div>For <span><math><mi>B</mi><mo>⊂</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>}</mo></math></span> and <span><math><mi>S</mi><mo>⊂</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, let <span><math><mrow><mi>Ann</mi></mrow><mo>(</mo><mi>B</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>β</mi><mo>∈</mo><mi>B</mi><mo>:</mo><mo>(</mo><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>)</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>}</mo></math></span>. Considering <span><math><mi>B</mi></math></span> also as a binary matrix with <span><math><mo>|</mo><mi>B</mi><mo>|</mo></math></span> rows and <em>n</em> columns, let <span><math><mi>r</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> denote the binary rank of <span><math><mi>B</mi></math></span>. We disprove here the conjecture of Stevanović [Linear Algebra Appl. 311 (2000) 35–44] that the components of NEPS of connected, bipartite graphs are almost cospectral whenever the basis <span><math><mi>B</mi></math></span> of NEPS satisfies the condition <span><math><mrow><mi>Ann</mi></mrow><mo>(</mo><mi>B</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>≠</mo><mo>∅</mo><mspace></mspace><mo>⇒</mo><mspace></mspace><mo>|</mo><mi>S</mi><mo>|</mo><mo>+</mo><mi>r</mi><mo>(</mo><mrow><mi>Ann</mi></mrow><mo>(</mo><mi>B</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mi>r</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"732 ","pages":"Pages 18-25"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682610","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":"Gershgorin-type spectral inclusions for matrices","authors":"Simon N. Chandler-Wilde ,&nbsp;Marko Lindner","doi":"10.1016/j.laa.2025.11.017","DOIUrl":"10.1016/j.laa.2025.11.017","url":null,"abstract":"<div><div>In this paper we derive sequences of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show for the example of large Toeplitz matrices. Our inclusion sets, which take the form of unions of pseudospectra of square or rectangular submatrices, build on our own recent work on inclusion sets for bi-infinite matrices in Chandler-Wilde et al. (2024) <span><span>[3]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"732 ","pages":"Pages 33-73"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682614","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":"Total trades, intersection matrices and Specht modules","authors":"Mihalis Maliakas,&nbsp;Dimitra-Dionysia Stergiopoulou","doi":"10.1016/j.laa.2025.11.021","DOIUrl":"10.1016/j.laa.2025.11.021","url":null,"abstract":"<div><div>Trades are important objects in combinatorial design theory that may be realized as certain elements of kernels of inclusion matrices. Total trades were introduced recently by Ghorbani, Kamali and Khosravshahi who showed that over a field of characteristic zero the vector space of trades decomposes into a direct sum of spaces of total trades. In this paper, we show that the vector space spanned by the permutations of a total trade is an irreducible representation of the symmetric group. As a corollary, the previous decomposition theorem is recovered. Also, a basis is obtained for the module of total trades in the spirit of Specht polynomials. In the second part of the paper we consider more generally intersection matrices and determine the irreducible decompositions of their images. This generalizes previously known results concerning ranks of special cases.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"732 ","pages":"Pages 74-92"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682615","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}