[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83009-en":3,"doc-seo-83009-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},83009,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","A Unified Framework for Formalizing Matrix Decomposition Proofs","Existence proofs for many matrix decompositions follow a common recursive pattern: normalize locally, select a matrix slice, solve a smaller subproblem recursively, then reconstruct by lifting through block structure. Formalizing this uniformly in dependent type theory is hard because recursion can alter index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. The work presents a Lean 4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting/transport, and subtype induction, and drives reusable proofs across PLU, LU, LDL/Cholesky, QR variants, SVD, and more.","arXiv :2607 .05874v1 [math .NA] 7 Jul 2026  \nA Unified Framework for Formalizing Matrix Decomposition  \nProofs  \nWanli Ma 1,* Zichen Wang2,*,† Zaiwen Wen 1  \n1 Beijing International Center for Mathematical Research, Peking University  \n2 School of Mathematical Sciences, Peking University  \nEmails: [wlma@pku.edu.cn](wlma@pku.edu.cn); [zichenwang25@stu.pku.edu.cn](zichenwang25@stu.pku.edu.cn); [wenzw@pku.edu.cn](wenzw@pku.edu.cn).  \nAbstract  \nExistence proofs for many matrix decompositions share a recursive routine: a local transformation prepares the matrix, a slice is selected, a recursive solution is obtained, and the result is lifted and transported back. Formalizing this routine uniformly in dependent type theory is difficult because recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. We develop a Lean 4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting, transport, and subtype induction. The framework uses general index types, packages square and rectangular matrices in universe types, and provides a decomposition driver that assembles strategy data into subtype-induction instances. It has been instantiated across PLU, LU, LDL/Cholesky, QR variants, Gauss rank normal form, Hessenberg reductions, Schur variants, normal spectral decomposition, SVD, bidiagonalization, tridiagonalization, UTV, Smith normal form, rational canonical form, and Jordan-type forms at varying levels of statement strength. Across these instances, repeated decomposition proofs are best treated not as separate tasks but as instances of a more general inductive statement whose interface records a certified proof path compatible with the chosen decomposition statement.  \n1 Introduction  \nMatrix decompositions are fundamental in linear algebra and scientific computing. They underlie the solution of linear systems, least-squares problems, eigenvalue computations, and lowrank approximation, and they provide the structural basis for many classical numerical algorithms [1, 2, 3, 4, 5, 6] . Lean 4 and its mathematical library mathlib provide substantial foundations for algebra, analysis, topology, and linear algebra [7, 8, 9], making matrix decompositions a natural target for formalization. Despite their different concrete forms, existence proofs for these decompositions often share a compact recursive shape: apply a local normalization, reduce to a smaller subproblem, solve it recursively, and lift the result back through block structure [1, 2, 3] . Similar  \nproof structures appear in matrix reduction procedures, constructions of normal forms, and related ∗ These authors contributed equally to this work.  \n†Corresponding author.  \nresults in linear algebra [1, 3, 10, 11, 12, 13, 14] . The recurrence of this pattern suggests that the common proof organization itself can be represented as formal structure.  \nRelated formalization projects have already shown that substantial mathematical results can be developed rigorously in proof assistants [15, 16, 17, 18, 19, 20, 21, 22] . In linear algebra, existing developments include formalizations of matrix echelon forms, Smith normal form in both Isabelle/HOLand Coq, and Jordan normal form with spectral radius theory [13, 14, 23, 24] . Further work covers QR decomposition [25] and the fundamental theorem of linear algebra [26] . Large-scale projects such as the formalization of the Feit-Thompson theorem [27] have required building substantial linear algebra libraries as infrastructure. For example, Aransay and Divasón formalized the computation of matrix echelon forms in Isabelle/HOL and proved the correctness of the corresponding algorithm [13] . Divasón and Thiemann later developed a systematic formalization of the Smith normal form, establishing the correctness of the relevant algorithm in a more general algebraic setting [14] . These developments show that classical result","cbCaip31KTZm9vzI","https://ap.wps.com/l/cbCaip31KTZm9vzI","pdf",761094,1,34,"English","en",105,"# Introduction\n## Unified recursive proof pattern\n## Motivation from existing formalizations\n# Framework architecture\n## Separation of schemas, strategies, and induction\n## Lifting, transport, and subtype induction\n# Instantiations across decompositions\n## PLU/LU/LDL-Cholesky and QR variants\n## Normal forms and spectral/SVD decompositions\n# Scope and implications","[{\"question\":\"What recurring structure do the paper’s matrix decomposition existence proofs share?\",\"answer\":\"They use a head-tail recursive routine: a local transformation normalizes the matrix, a slice is chosen, a smaller subproblem is solved recursively, and the result is lifted back through the surrounding block structure.\"},{\"question\":\"Why is it difficult to formalize this pattern uniformly in dependent type theory?\",\"answer\":\"Recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings, which requires careful transport and induction at the type and predicate levels.\"},{\"question\":\"What does the Lean 4 framework contribute to reuse decomposition proofs?\",\"answer\":\"It separates decomposition schemas, transformations, reduction strategies, measures, lifting/transport, and subtype induction, and provides a decomposition driver that assembles strategy data into subtype-induction instances so repeated proofs become instances of a more general inductive statement.\"}]",1784184636,86,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"a-unified-framework-for-formalizing-matrix-decomposition-proofs","",{"@graph":35,"@context":84},[36,53,67],{"@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/a-unified-framework-for-formalizing-matrix-decomposition-proofs/83009/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What recurring structure do the paper’s matrix decomposition existence proofs share?","Question",{"text":74,"@type":75},"They use a head-tail recursive routine: a local transformation normalizes the matrix, a slice is chosen, a smaller subproblem is solved recursively, and the result is lifted back through the surrounding block structure.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why is it difficult to formalize this pattern uniformly in dependent type theory?",{"text":79,"@type":75},"Recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings, which requires careful transport and induction at the type and predicate levels.",{"name":81,"@type":72,"acceptedAnswer":82},"What does the Lean 4 framework contribute to reuse decomposition proofs?",{"text":83,"@type":75},"It separates decomposition schemas, transformations, reduction strategies, measures, lifting/transport, and subtype induction, and 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