{"paper":{"title":"The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Finite-rank normal perturbations create outlier eigenvalues outside the bulk spectrum of large rotationally invariant non-Hermitian random matrices.","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.dis-nn","authors_text":"Marc Potters, Pierre Bousseyroux","submitted_at":"2026-01-15T14:21:54Z","abstract_excerpt":"We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-P\\'ech\\'e (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\\mathbf{A} + \\mathbf{T}$, where $\\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize the emergence and fluctuations of outlier eigenvalues in models of the form A + T, where A is a large rotationally invariant non-Hermitian random matrix and T is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unperturbed matrix A must be rotationally invariant and the perturbation T must be normal and of finite rank; if either condition fails the outlier formulas no longer hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finite-rank normal perturbations of large rotationally invariant non-Hermitian random matrices produce outlier eigenvalues whose positions and fluctuations, together with the associated eigenvectors, are characterized in a unified framework that includes the Hermitian case.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite-rank normal perturbations create outlier eigenvalues outside the bulk spectrum of large rotationally invariant non-Hermitian random matrices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b2bde0325145da6bd954041dcb816ccb40d5b0c7a4b1b15d5442d07d391f8fb4"},"source":{"id":"2601.10427","kind":"arxiv","version":2},"verdict":{"id":"30611388-d8f0-45f9-8b1a-897147110a3a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:11:27.613458Z","strongest_claim":"We characterize the emergence and fluctuations of outlier eigenvalues in models of the form A + T, where A is a large rotationally invariant non-Hermitian random matrix and T is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior.","one_line_summary":"Finite-rank normal perturbations of large rotationally invariant non-Hermitian random matrices produce outlier eigenvalues whose positions and fluctuations, together with the associated eigenvectors, are characterized in a unified framework that includes the Hermitian case.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unperturbed matrix A must be rotationally invariant and the perturbation T must be normal and of finite rank; if either condition fails the outlier formulas no longer hold.","pith_extraction_headline":"Finite-rank normal perturbations create outlier eigenvalues outside the bulk spectrum of large rotationally invariant non-Hermitian random matrices."},"references":{"count":20,"sample":[{"doi":"","year":2008,"title":"Central limit theorems for eigenvalues in a spiked population model.Ann","work_id":"a3d7c15e-5058-4d79-831c-af1033537e84","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.Ann","work_id":"88dddc9a-00c8-4a56-916d-5e5d91c7ee2c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Belinschi, Hari Bercovici, Mireille Capitaine, and Maxime F´ evrier","work_id":"3f9b6b42-0d40-455f-bb83-a325fa02b3ee","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices.Advances in Mathematics, 227(1):494–521","work_id":"a6eaf4f9-bd2a-4e4d-b652-1fbb1a4bd539","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Outliers in the single ring theorem.Probab","work_id":"8d7e8e13-ad1b-42a4-bc5a-45f35a04b7e2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"9b2e881a2f5fc58e0218fa13d7f6e6a069961136e7a1386c02935d8f1dd6805d","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"154322d07f06f22cadef03ad2a3ac0aec749e5644c10d9b5408770b377a8d180"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}