{"title":"一个逆三对角特征值问题及其在网络运动同步中的应用","authors":"Luca Dieci , Cinzia Elia , Alessandro Pugliese","doi":"10.1016/j.laa.2025.08.015","DOIUrl":null,"url":null,"abstract":"<div><div>In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and null-vector <figure><img></figure>. Then, we show how this result can be used to guarantee –if possible– that a synchronous orbit of a connected tridiagonal network associated to the matrix <em>L</em> above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for <em>L</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 26-46"},"PeriodicalIF":1.1000,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion\",\"authors\":\"Luca Dieci , Cinzia Elia , Alessandro Pugliese\",\"doi\":\"10.1016/j.laa.2025.08.015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi></mrow></msub></math></span> and null-vector <figure><img></figure>. Then, we show how this result can be used to guarantee –if possible– that a synchronous orbit of a connected tridiagonal network associated to the matrix <em>L</em> above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for <em>L</em>.</div></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":\"728 \",\"pages\":\"Pages 26-46\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2025-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379525003581\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525003581","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion
In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues and null-vector . Then, we show how this result can be used to guarantee –if possible– that a synchronous orbit of a connected tridiagonal network associated to the matrix L above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for L.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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