Pub Date : 2026-01-26DOI: 10.1016/j.laa.2026.01.027
Fredy Yip
The sum of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer , let denote the supremum of over graphs without odd cycles of length less than k. The example of the k-cycle shows that . In their recent work, Abiad, Taranchuk, and Van Veluw showed that and asked to determine the asymptotics of . Using approximation theory, we show that , giving a tight upper bound up to a poly-logarithmic factor.
{"title":"On the sum of the largest and smallest eigenvalues of graphs with high odd girth","authors":"Fredy Yip","doi":"10.1016/j.laa.2026.01.027","DOIUrl":"10.1016/j.laa.2026.01.027","url":null,"abstract":"<div><div>The sum <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, let <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denote the supremum of <span><math><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></math></span> over graphs without odd cycles of length less than <em>k</em>. The example of the <em>k</em>-cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> shows that <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≥</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span>. In their recent work, Abiad, Taranchuk, and Van Veluw showed that <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> and asked to determine the asymptotics of <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. Using approximation theory, we show that <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>k</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>, giving a tight upper bound up to a poly-logarithmic factor.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"736 ","pages":"Pages 1-12"},"PeriodicalIF":1.1,"publicationDate":"2026-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070978","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 : 2026-01-26DOI: 10.1016/j.laa.2026.01.023
Amir Rehman, S. Pirzada
Let G be a graph of size m and be the spectral radius of its adjacency matrix. Let be the graph obtained from k triangles and a quadrilateral sharing a common vertex. The graph is also called the fish graph. Let be the graph obtained by joining each vertex of to isolated vertices. Li and Peng (2022) solved the spectral Turán problem for intersecting odd cycles and Desai (2024) determined the unique graph with maximum spectral radius among all graphs that do not contain intersecting even cycles, for n sufficiently large. Further, Li, Lu and Peng (2023) determined the unique m-edge graph with maximum spectral radius forbidding two intersecting triangles at a common vertex. Now, a natural question that arises is determining the maximum spectral radius for graphs that avoid intersecting cycles, with at least one of odd length and at least one of even length. In this paper, we address this question by investigating the spectral extremal problem for the graph formed by the intersection of an even cycle and an odd cycle at a common vertex. We show that if G is an -free graph of size , then , with equality if and only if .
{"title":"Spectral extremal problem for intersecting one even and one odd cycle","authors":"Amir Rehman, S. Pirzada","doi":"10.1016/j.laa.2026.01.023","DOIUrl":"10.1016/j.laa.2026.01.023","url":null,"abstract":"<div><div>Let <em>G</em> be a graph of size <em>m</em> and <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the spectral radius of its adjacency matrix. Let <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the graph obtained from <em>k</em> triangles and a quadrilateral sharing a common vertex. The graph <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is also called the fish graph. Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span> be the graph obtained by joining each vertex of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> to <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> isolated vertices. Li and Peng (2022) solved the spectral Turán problem for intersecting odd cycles and Desai (2024) determined the unique graph with maximum spectral radius among all graphs that do not contain intersecting even cycles, for <em>n</em> sufficiently large. Further, Li, Lu and Peng (2023) determined the unique <em>m</em>-edge graph with maximum spectral radius forbidding two intersecting triangles at a common vertex. Now, a natural question that arises is determining the maximum spectral radius for graphs that avoid intersecting cycles, with at least one of odd length and at least one of even length. In this paper, we address this question by investigating the spectral extremal problem for the graph formed by the intersection of an even cycle and an odd cycle at a common vertex. We show that if <em>G</em> is an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-free graph of size <span><math><mi>m</mi><mo>≥</mo><mn>44</mn></math></span>, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, with equality if and only if <span><math><mi>G</mi><mo>≅</mo><msub><mrow><mi>S</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"736 ","pages":"Pages 45-53"},"PeriodicalIF":1.1,"publicationDate":"2026-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070980","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 : 2026-01-23DOI: 10.1016/j.laa.2026.01.022
Liwen Zhang, Zhiyuan Zhang
Let be a set of graphs on the same vertex set where is even. We say admits a rainbow k-factor if there exists a k-regular graph F on the vertex set V such that all edges of F are from different members of . In this paper, we show a sufficient spectral condition for the existence of a rainbow k-factor for , which is that if for each , then admits a rainbow k-factor unless .
{"title":"Spectral radius and rainbow k-factors of graphs","authors":"Liwen Zhang, Zhiyuan Zhang","doi":"10.1016/j.laa.2026.01.022","DOIUrl":"10.1016/j.laa.2026.01.022","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>}</mo></math></span> be a set of graphs on the same vertex set <span><math><mi>V</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> where <span><math><mi>k</mi><mo>⋅</mo><mi>n</mi></math></span> is even. We say <span><math><mi>G</mi></math></span> admits a rainbow <em>k</em>-factor if there exists a <em>k</em>-regular graph <em>F</em> on the vertex set <em>V</em> such that all edges of <em>F</em> are from different members of <span><math><mi>G</mi></math></span>. In this paper, we show a sufficient spectral condition for the existence of a rainbow <em>k</em>-factor for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, which is that if <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>ρ</mi><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> for each <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>G</mi></math></span>, then <span><math><mi>G</mi></math></span> admits a rainbow <em>k</em>-factor unless <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 307-318"},"PeriodicalIF":1.1,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090724","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}
Completely positive trace preserving maps are widely used in quantum information theory. These are mostly studied using the master equation perspective. A central part in this theory is to study whether a given system of dynamical maps is Markovian or non-Markovian. We study the problem when the underlying Hilbert space is of infinite dimension. We construct a sufficient condition for checking P (resp. CP) divisibility of dynamical maps. We construct several examples where the underlying Hilbert space may not be of finite dimension. We also give a special emphasis on Gaussian dynamical maps and get a version of our result in it.
{"title":"Divisibility of infinite dimensional dynamical maps","authors":"Bihalan Bhattacharya , Uwe Franz , Saikat Patra , Ritabrata Sengupta","doi":"10.1016/j.laa.2026.01.020","DOIUrl":"10.1016/j.laa.2026.01.020","url":null,"abstract":"<div><div>Completely positive trace preserving maps are widely used in quantum information theory. These are mostly studied using the master equation perspective. A central part in this theory is to study whether a given system of dynamical maps <span><math><mo>{</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>:</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></math></span> is Markovian or non-Markovian. We study the problem when the underlying Hilbert space is of infinite dimension. We construct a sufficient condition for checking P (resp. CP) divisibility of dynamical maps. We construct several examples where the underlying Hilbert space may not be of finite dimension. We also give a special emphasis on Gaussian dynamical maps and get a version of our result in it.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"736 ","pages":"Pages 13-44"},"PeriodicalIF":1.1,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070979","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 : 2026-01-20DOI: 10.1016/j.laa.2026.01.019
Qiang Dong , Shunye Li
In this article, we present quiver realizations of two classes of algebras that are derived equivalent to upper triangular matrix algebras. We investigate defective rectangle algebras and show that four of them are derived equivalent. Moreover, we establish derived equivalences between certain defective rectangle algebras and Nakayama algebras.
{"title":"Derived equivalences between defective rectangles","authors":"Qiang Dong , Shunye Li","doi":"10.1016/j.laa.2026.01.019","DOIUrl":"10.1016/j.laa.2026.01.019","url":null,"abstract":"<div><div>In this article, we present quiver realizations of two classes of algebras that are derived equivalent to upper triangular matrix algebras. We investigate defective rectangle algebras and show that four of them are derived equivalent. Moreover, we establish derived equivalences between certain defective rectangle algebras and Nakayama algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 287-306"},"PeriodicalIF":1.1,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036323","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 : 2026-01-15DOI: 10.1016/j.laa.2026.01.014
James Demmel , Ioana Dumitriu , Ryan Schneider
This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil , in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to this problem that combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function.
{"title":"Fast and inverse-free algorithms for deflating subspaces","authors":"James Demmel , Ioana Dumitriu , Ryan Schneider","doi":"10.1016/j.laa.2026.01.014","DOIUrl":"10.1016/j.laa.2026.01.014","url":null,"abstract":"<div><div>This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on <em>spectral</em> projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to this problem that combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 222-258"},"PeriodicalIF":1.1,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036319","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 : 2026-01-15DOI: 10.1016/j.laa.2026.01.008
Ralph Morrison
We study the commuting graph of matrices over the field of p-adics , whose vertices are non-scalar matrices with entries in and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if , with n neither prime nor a power of p. We also prove that in the case of and for q a prime with , the commuting graph has the maximum possible diameter of 6; these are the first known such examples independent of the axiom of choice. We also find choices of p and n yielding diameter 4 and diameter 5 commuting graphs, and prove general bounds depending on p and n.
{"title":"Commuting graphs of p-adic matrices","authors":"Ralph Morrison","doi":"10.1016/j.laa.2026.01.008","DOIUrl":"10.1016/j.laa.2026.01.008","url":null,"abstract":"<div><div>We study the commuting graph of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over the field of <em>p</em>-adics <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, whose vertices are non-scalar <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices with entries in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, with <em>n</em> neither prime nor a power of <em>p</em>. We also prove that in the case of <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>q</mi></math></span> for <em>q</em> a prime with <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span>, the commuting graph has the maximum possible diameter of 6; these are the first known such examples independent of the axiom of choice. We also find choices of <em>p</em> and <em>n</em> yielding diameter 4 and diameter 5 commuting graphs, and prove general bounds depending on <em>p</em> and <em>n</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 188-202"},"PeriodicalIF":1.1,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036317","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 : 2026-01-14DOI: 10.1016/j.laa.2026.01.009
Daryl Q. Granario
A complex matrix A is said to be coninvolutory if A is nonsingular and its inverse is its conjugate . We show that every complex matrix A with is a product of four coninvolutory matrices. We also characterize matrices that are products of three coninvolutory matrices. Finally, we give a concanonical form characterization for matrices that are sums of two coninvolutory matrices.
{"title":"On products and sums of coninvolutory matrices","authors":"Daryl Q. Granario","doi":"10.1016/j.laa.2026.01.009","DOIUrl":"10.1016/j.laa.2026.01.009","url":null,"abstract":"<div><div>A complex matrix <em>A</em> is said to be coninvolutory if <em>A</em> is nonsingular and its inverse is its conjugate <span><math><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></math></span>. We show that every complex matrix <em>A</em> with <span><math><mo>|</mo><mi>det</mi><mo></mo><mi>A</mi><mo>|</mo><mo>=</mo><mn>1</mn></math></span> is a product of four coninvolutory matrices. We also characterize matrices that are products of three coninvolutory matrices. Finally, we give a concanonical form characterization for matrices that are sums of two coninvolutory matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 175-187"},"PeriodicalIF":1.1,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036364","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 : 2026-01-14DOI: 10.1016/j.laa.2026.01.011
Daniel Kressner , David Persson , André Uschmajew
Randomized methods, such as the randomized SVD (singular value decomposition) and Nyström approximation, are an effective way to compute low-rank approximations of large matrices. Motivated by applications to operator learning, Boullé and Townsend (FoCM, 2023) recently proposed an infinite-dimensional extension of the randomized SVD for a Hilbert–Schmidt operator that invokes randomness through a Gaussian process with a covariance operator . While the non-isotropy introduced by allows one to incorporate prior information on , an unfortunate choice may lead to unfavorable performance and large constants in the error bounds. In this work, we introduce a novel infinite-dimensional extension of the randomized SVD that does not require such a choice and enjoys error bounds that match those for the finite-dimensional case. Our extension implicitly uses isotropic random vectors, reflecting a choice commonly made in the finite-dimensional case. In fact, the theoretical results of this work show how the usual randomized SVD applied to a discretization of approaches our infinite-dimensional extension as the discretization gets refined, both in terms of error bounds and the Wasserstein distance. We also present and analyze a novel extension of the Nyström approximation for self-adjoint positive semi-definite trace class operators.
{"title":"On the randomized SVD in infinite dimensions","authors":"Daniel Kressner , David Persson , André Uschmajew","doi":"10.1016/j.laa.2026.01.011","DOIUrl":"10.1016/j.laa.2026.01.011","url":null,"abstract":"<div><div>Randomized methods, such as the randomized SVD (singular value decomposition) and Nyström approximation, are an effective way to compute low-rank approximations of large matrices. Motivated by applications to operator learning, Boullé and Townsend (FoCM, 2023) recently proposed an infinite-dimensional extension of the randomized SVD for a Hilbert–Schmidt operator <span><math><mi>A</mi></math></span> that invokes randomness through a Gaussian process with a covariance operator <span><math><mi>K</mi></math></span>. While the non-isotropy introduced by <span><math><mi>K</mi></math></span> allows one to incorporate prior information on <span><math><mi>A</mi></math></span>, an unfortunate choice may lead to unfavorable performance and large constants in the error bounds. In this work, we introduce a novel infinite-dimensional extension of the randomized SVD that does not require such a choice and enjoys error bounds that match those for the finite-dimensional case. Our extension implicitly uses isotropic random vectors, reflecting a choice commonly made in the finite-dimensional case. In fact, the theoretical results of this work show how the usual randomized SVD applied to a discretization of <span><math><mi>A</mi></math></span> approaches our infinite-dimensional extension as the discretization gets refined, both in terms of error bounds and the Wasserstein distance. We also present and analyze a novel extension of the Nyström approximation for self-adjoint positive semi-definite trace class operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 259-286"},"PeriodicalIF":1.1,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036320","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 : 2026-01-14DOI: 10.1016/j.laa.2026.01.013
John Bamberg , Jesse Lansdown
We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most 8191 is a I-group. We also consider design-orthogonality more generally for noncommutative homogeneous coherent configurations.
{"title":"The synchronisation hierarchy via coherent configurations","authors":"John Bamberg , Jesse Lansdown","doi":"10.1016/j.laa.2026.01.013","DOIUrl":"10.1016/j.laa.2026.01.013","url":null,"abstract":"<div><div>We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most 8191 is a <span><math><mi>Q</mi></math></span>I-group. We also consider design-orthogonality more generally for noncommutative homogeneous coherent configurations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"735 ","pages":"Pages 203-221"},"PeriodicalIF":1.1,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146036318","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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