[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85407-en":3,"doc-seo-85407-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85407,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Matrix Nearness Problems And Eigenvalue Optimization","A matrix nearness problem seeks, for a given matrix A, the closest matrix within a prescribed class under a matrix-norm measure. The work focuses on nearness tasks tied to eigenvalues, singular values, and pseudospectra, motivated by robust stability/control and graph clustering/ranking. It characterizes optimal perturbations as rank-one matrices or projections of rank-one matrices onto structured linear constraints, computed via nested or alternating two-stage iterations combining rank-one eigenvalue optimization through gradient-based differential equations with scalar nonlinear root-finding.","arXiv :2503 . 14750v4 [math .NA] 13 Jul 2026  \nMatrix nearness problems and eigenvalue optimization  \nNicola Guglielmi and Christian Lubich  \nPreprint version  \nJuly 14, 2026  \nDedicated to our children.  \nvi Preface  \nPreface  \nA matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured ina matrix norm. (Nicholas J. Higham, 1989)  \nThis book is about solving matrix nearness problems that are related to eigenvalues or singular values or pseudospectra. These problems arise in great diversity in various fields, be they related to dynamics, as in questions of robust stability and robust control, or related to graphs, as in questions of clustering and ranking. Algorithms for such problems work with matrix perturbations that drive eigenvalues or singular values or Rayleigh quotientsto desired locations.  \nRemarkably, the optimal perturbation matrices are typically of rank one or are projections of rank-1 matrices onto a linear structure, e.g. a prescribed sparsity pattern. In the approach worked out here, these optimal rank-1 perturbations will be determined ina two-level iteration that consists of eigenvalue optimization and root-finding. An eigenvalue optimization problem with equality or inequality constraints on the perturbation size is to be solved via gradient-based rank-1 matrix differential equations. This amounts to numerically driving a rank-1 matrix, which is represented by two vectors, to a stationary point. The root-finding part determines the optimal perturbation size by solving a scalar nonlinear equation. The two algorithmic parts can be nested or can alternate.  \nA wide variety of matrix nearness problems, as outlined in the introductory Chapter I, will be tackled by such an approach and its nontrivial extensions. In Chapter II, we study a basic eigenvalue optimization problem and its numerical solution via rank-1 matrix differential equations, which are norm-and rank-1 constrained gradient systems. In Chapter III, this approach yields algorithms for computing extremal points and boundary curves of pseudospectra. In Chapter IV, we present algorithms for matrix nearness problems, which in a nested or alternating way combine the rank-1 eigenvalue optimization algorithms of Chapter II with a Newton–bisection method. This is illustrated by the problem of computing a nearest unstable complex matrix to a given stable matrix. In Chapter V, the rank-1 approach is extended to nearness problems for matrices with a prescribed complex-linear or real-linear structure, e.g. real matrices or matrices with a given sparsity pattern or a Toeplitz or Hankel structure. The approach is applied to various structured matrix nearness problems, which include finding the nearest singular structured matrix toa given invertible structured matrix and, for asymptotically stable linear differential equations, finding the largest norm of structured perturbations of the matrix that still ensuresa prescribed transient bound. In Chapter VI, we propose and analyze algorithms for matrix nearness problems that go beyond those of Chapters IV and V, among them matrix stabilization and finding a nearest defective real or complex matrix. In Chapter VII, we discuss algorithms for exemplary nearness problems in the area of systems and control, and in Chapter VIII we rephrase clustering and ranking problems from graph theory as structured matrix nearness problems and extend our algorithmic approach to such graph problems.  \nPreface vii  \nIn Chapters II to VIII, the inclusion of references to the existing literature is done in Notes at the end of each chapter, and only exceptionally we add references to the running text. References are not numbered, but are addressed by the names of authors and the year of publication, for example Higham (1989) and Lewis & Overton (1996) .  \nMATLAB codes implementing the algorithms presented in this book are freely available for non-comm","cbCaikGyFbw3Y5pf","https://ap.wps.com/l/cbCaikGyFbw3Y5pf","pdf",2051263,1,265,"English","en",105,"# Introduction by examples\n# A basic eigenvalue optimization problem\n## Problem formulation\n## Gradient flow\n## Rank-1 constrained gradient flow\n# Extremal points and boundary curves of pseudospectra\n# Structured matrix nearness problems\n# Matrix stabilization and nearest defective matrices\n# Systems and control applications\n# Graph clustering and ranking as matrix nearness","[{\"question\":\"What does a matrix nearness problem aim to compute?\",\"answer\":\"It finds, for an arbitrary matrix A, the nearest matrix in a specified class according to a distance measured in a matrix norm.\"},{\"question\":\"How are optimal perturbations characterized in this approach?\",\"answer\":\"Optimal perturbation matrices are typically rank one, or projections of rank-one matrices onto a linear structure such as a prescribed sparsity pattern.\"},{\"question\":\"What is the core algorithmic strategy for the eigenvalue-related nearness problems?\",\"answer\":\"It uses a two-level iteration: rank-one eigenvalue optimization via gradient-based rank-one matrix differential equations, and root-finding using a scalar nonlinear equation for the optimal perturbation size.\"}]",1784203173,668,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"matrix-nearness-problems-and-eigenvalue-optimization","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/matrix-nearness-problems-and-eigenvalue-optimization/85407/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does a matrix nearness problem aim to compute?","Question",{"text":75,"@type":76},"It finds, for an arbitrary matrix A, the nearest matrix in a specified class according to a distance measured in a matrix norm.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are optimal perturbations characterized in this approach?",{"text":80,"@type":76},"Optimal perturbation matrices are typically rank one, or projections of rank-one matrices onto a linear structure such as a prescribed sparsity pattern.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the core algorithmic strategy for the eigenvalue-related nearness problems?",{"text":84,"@type":76},"It uses a two-level iteration: rank-one eigenvalue optimization via gradient-based rank-one matrix differential equations, and root-finding using a scalar nonlinear equation for the optimal perturbation 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