[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83089-en":3,"doc-seo-83089-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83089,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Fast Computation of Eigenvalues of Periodic CMV Matrices","Periodic CMV matrices are unitary matrices specified by O(n) parameters. Their eigenvalues can be computed using conventional storage in O(n^2) memory and O(n^3) time, but a faster approach is needed. The paper presents an algorithm that computes eigenvalues of periodic CMV (Floquet) matrices in O(n^2) time while using O(n) data. The method exploits structured core transformations and periodicity of the underlying Verblunsky coefficients.","FAST COMPUTATION OF EIGENVALUES OF PERIODIC CMV  \nMATRICES  \nTHOMAS MACH∗ , MICHELE RINELLI†, RAF VANDEBRIL†, AND DAVID S. WATKINS‡  \nAbstract. Periodic CMV matrices are unitary matrices that can be specified by O (n) data. Their eigenvalues can be computed by standard methods, storing them as conventional matrices (using O (n2 ) data) in O (n3 ) time. Here a fast method that computes the eigenvalues in O (n2 ) time (using O (n) data)  \nis presented.  \narXiv :2607 .06400v1 [math .NA] 7 Jul 2026  \n1. Introduction  \nThe paper [3] by Damanik, Embree, and Fillman contains an interesting unitary eigenvalue problem. The starting point is an infinite CMV matrix. If a certain periodicity condition is satisfied, a related finite matrix can be extracted. This periodic CMV matrix (or Floquet matrix) is an n × n (with even n) unitary matrix that is determined by O (n) data, and its eigenvalues are of interest [3, Prop. 2.5] .(The eigenvectors are not needed.) Certainly the problem can be solved by storing it in a conventional matrix format using O(n2 ) data, and using standard software to compute the eigenvalues in O (n3 ) time. However, one would hope to compute eigenvalues in O(n2 ) time, given that the matrix can be represented by O(n) data. This is indeed possible, and we will show how to do it.  \nAn infinite CMV matrix is a unitary matrix that is a product of two very simple block-diagonal matrices  \n􀀲  \n􀀶  \n􀀶  \n􀀶  \nL = 􀀶  \n􀀶  \n􀀶  \n􀀶  \n􀀴  \n.  \n.  \n.  \nC0 C2  \nC4  \n.  \n.  \n.  \n􀀳  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀵  \n2020 Mathematics Subject Classification. 65F15, 65H17, 15A18, 65H04 .  \nKey words and phrases. unitary matrix, CMV matrix, Floquet matrix, eigenvalue, fast algorithm.  \n∗ Institute for Mathematics, University of Potsdam, [Germany.](Germany. thomas.mach@gmail.com)[ thomas.mach@gmail.com](Germany. thomas.mach@gmail.com).  \n†Numerical Analysis and Applied Mathematics (NUMA) unit, Department of Computer Science, KU Leuven, Leuven, [Belgium.](Belgium. michele.rinelli@kuleuven.be)[ michele.rinelli@kuleuven.be](Belgium. michele.rinelli@kuleuven.be),  \n[raf.vandebril@kuleuven.be](raf.vandebril@kuleuven.be).  \n‡Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA. [davidwatkinsmatrix@gmail.com](davidwatkinsmatrix@gmail.com).  \n2 T. MACH, M. RINELLI AND R. VANDEBRIL, AND D. S. WATKINS  \nand  \n􀀲􀀶􀀶􀀶  \nM = 􀀶􀀶􀀶􀀶􀀴  \n.  \n.  \n.  \nC1 C3  \nC5  \n􀀳  \n􀀷  \n􀀷  \n􀀷  \n􀀷  \n􀀷 .  \n􀀷  \n􀀷  \n􀀷  \n. . . 􀀵  \nEach of these matrices is infinite in both directions. Each block Ci is a 2×2 unitary matrix, so it could be, for example, a rotator or a reflector. The blocks in L and Mare not aligned. For example, the first row (column) of C1 is even with the second row (column) of C0 , and the second row (column) of C1 is even with the first row (column) of C2 . The product LM, which is known as a CMV matrix, is thus not block diagonal but pentadiagonal with some special structure. We will not display this because we are going to keep the matrix in its factored form.  \nNow suppose the matrix entries satisfy a periodicity condition Ck+n = Ck , where n is even. Then all of the information is contained in a finite submatrix built from C1 , C3 , ... , Cn−1 and C2 , C4 , ... , Cn , for example, and we can build an n × n periodic CMV matrix, the eigenvalues of which are of interest [3, Prop. 2.5] .  \n2. Core transformations  \nBefore we can proceed, we need to introduce some notation and terminology. We’ll keep it brief; for more details see, for example,[1] .  \nA core transformation, briefly core, is a unitary matrix that equals the identity matrix except in two consecutive rows/columns. Core transformations have lots of uses; we will start by looking at the reduction of an upper Hessenberg matrix  \nH =  \n××××××  \n××××××  \n××××××××××××××  \nto upper triangular form. First a core transformation acts on rows 1 and 2 of H to transform the (2 , 1) entry to zero. This operation is depicted by  \n↱↱  \n\n| ××××××\u003Cbr>××××××\u003Cbr>×××××\u003Cbr>××××\u003Cbr>×××\u003Cbr>××","cbCaitZ0ZnKQ19cc","https://ap.wps.com/l/cbCaitZ0ZnKQ19cc","pdf",374961,3,1,13,"English","en",105,"# Introduction\n# Core transformations","[{\"question\":\"What kind of matrices are periodic CMV matrices, and how are they represented?\",\"answer\":\"Periodic CMV matrices are unitary matrices defined through O(n) data, under a periodicity condition on the underlying 2×2 block parameters.\"},{\"question\":\"How does the paper improve eigenvalue computation compared with standard methods?\",\"answer\":\"Standard methods store the matrix explicitly using O(n^2) data and compute eigenvalues in O(n^3) time. The paper gives an O(n^2)-time method using only O(n) data.\"},{\"question\":\"What are “core transformations” in the proposed approach?\",\"answer\":\"Core transformations are unitary matrices equal to the identity except for two consecutive rows/columns. The algorithm uses sequences of these operations to efficiently reduce related structured matrices without forming them explicitly.\"}]",1784185127,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"fast-computation-of-eigenvalues-of-periodic-cmv-matrices","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/fast-computation-of-eigenvalues-of-periodic-cmv-matrices/83089/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 kind of matrices are periodic CMV matrices, and how are they represented?","Question",{"text":75,"@type":76},"Periodic CMV matrices are unitary matrices defined through O(n) data, under a periodicity condition on the underlying 2×2 block parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper improve eigenvalue computation compared with standard methods?",{"text":80,"@type":76},"Standard methods store the matrix explicitly using O(n^2) data and compute eigenvalues in O(n^3) time. The paper gives an O(n^2)-time method using only O(n) data.",{"name":82,"@type":73,"acceptedAnswer":83},"What are “core transformations” in the proposed approach?",{"text":84,"@type":76},"Core transformations are unitary matrices equal to the identity except for two consecutive rows/columns. The algorithm uses sequences of these operations to efficiently reduce related structured matrices without forming them explicitly.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]