{"paper":{"title":"Block Krylov subspaces and orthogonal matrix polynomials: a structural correspondence with applications to unitary matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree under a no-deflation assumption.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Michele Rinelli, Raf Vandebril","submitted_at":"2026-05-16T12:11:31Z","abstract_excerpt":"We study the connection between block Krylov subspaces and matrix orthogonal functions.\n  Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations. The same correspondence holds for rational block Krylov subspaces and matrix-valued rational functions, and in the extended Krylov setting this leads naturally to Laurent matrix polynomials.\n  When the matrix $A$ is normal, we prove that t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under a no-deflation assumption, polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations; for unitary matrices this transfers the Szegő recurrence and CMV framework to yield efficient orthogonalization procedures.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The no-deflation assumption on the block Krylov process, which is invoked to guarantee that the generated subspace has full dimension and that the isometric isomorphism to the matrix-polynomial space holds without breakdown (abstract, paragraph on polynomial block Krylov subspaces).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Block Krylov subspaces correspond isometrically to matrix polynomial spaces, allowing transfer of Szegő recurrences and CMV frameworks to orthogonalize polynomial and extended block Krylov bases for unitary matrices.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree under a no-deflation assumption.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0acd4dc28491ce0ed86aa7643c5f6372ec767f617ecc82b27f0b99eeb574faae"},"source":{"id":"2605.16954","kind":"arxiv","version":1},"verdict":{"id":"944884ae-f01f-498c-ac46-834ca70ab65d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:29:46.713301Z","strongest_claim":"Under a no-deflation assumption, polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations; for unitary matrices this transfers the Szegő recurrence and CMV framework to yield efficient orthogonalization procedures.","one_line_summary":"Block Krylov subspaces correspond isometrically to matrix polynomial spaces, allowing transfer of Szegő recurrences and CMV frameworks to orthogonalize polynomial and extended block Krylov bases for unitary matrices.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The no-deflation assumption on the block Krylov process, which is invoked to guarantee that the generated subspace has full dimension and that the isometric isomorphism to the matrix-polynomial space holds without breakdown (abstract, paragraph on polynomial block Krylov subspaces).","pith_extraction_headline":"Polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree under a no-deflation assumption."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16954/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:18.927828Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T19:51:58.864592Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:40:41.952705Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.235819Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.320124Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f20800945f34d2c17e1901d7de36888dd4de9e3ba1692cccdf6d0ceceeeac74a"},"references":{"count":27,"sample":[{"doi":"","year":2005,"title":"A. C. Antoulas.Approximation of large-scale dynamical systems, volume 6 ofAdvances in Design and Control. Society for Industrial and Applied Mathematics, 2005","work_id":"b41f5e4c-a362-42f4-80d2-23df1dc00e17","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"E. Carson, K. Lund, M. Rozloˇ zn´ ık, and S. Thomas. Block Gram-Schmidt algorithms and their stability proper- ties.Linear Algebra Appl., 638:150–195, 2022","work_id":"5df9564a-f935-4102-acc8-1f1254b3c762","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"D. Damanik, M. Embree, and J. Fillman. Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism.Indag. Math. (N.S.), 35(5):813–836, 2024","work_id":"f39d9262-85b1-41d4-b37a-92bf3d2df03f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"D. Damanik, A. Pushnitski, and B. Simon. The analytic theory of matrix orthogonal polynomials.Surv. Approx. Theory, 4:1–85, 2008","work_id":"a3cec6d9-b18e-4d28-a05f-2f522a3294bc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"V. Druskin and L. Knizhnerman. Extended Krylov subspaces: approximation of the matrix square root and related functions.SIAM J. Matrix Anal. Appl., 19(3):755–771, 1998","work_id":"c97b3a08-33e1-4dcd-ae51-f5473e36a3f2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":27,"snapshot_sha256":"a6665d1676e244968638fcf0738211fd6bd296b52277a85be3ddf67803221262","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"bf920d489a929611b71ac9818e825e59a4b99af43c113e0613821d53ef518a3c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}