{"corpus_id":122307911,"paper_sha":"3407282ac465137a16b2c6b0e3fad9d039de4de3","doi":"10.1016/0024-3795(93)00320-Y","arxiv_id":null,"pmid":null,"pmcid":null,"mag_id":2060886109,"dblp_id":null,"acl_id":null,"title":"The generalized spectral-radius theorem: An analytic-geometric proof","year":1995,"publication_date":"1995-04-15","venue":"","journal":{"name":"Linear Algebra and its Applications","pages":"151-159","volume":"220"},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":8,"citation_count":142,"influential_citation_count":13,"is_open_access":false,"arxiv_categories":null,"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Abstract Let ∑ be a bounded set of complex matrices, ∑ m = {A 1 ... A m : A i ∈ ∑} . The generalized spectral-radius theorem states that ϱ(∑) =ρ^(∑) , where ϱ(∑) and ρ^(σ) are defined as follows: ϱ{∑) =lim sup ϱ m (∑){1/m} , where ϱ m (∑) =sup{ϱ(A): A ∈ ∑ m } with ϱ ( A ) the spectral radius; ρ^(∑) =lim sup ρ^ m (∑){1/m} , where ρ^ m (∑) =sup{‖A‖: A ∈ ∑ m } with ‖ ‖ any matrix norm. We give an elementary proof, based on analytic and geometric tools, which is in some ways simpler than the first proof by Berger and Wang.","claims":[{"public_id":"cl_b20a239ac0dc6a5107de6d6c91e9df51","status":"active","text":"An elementary proof of the generalized spectral-radius theorem is given, based on analytic and geometric tools, which is in some ways simpler than the first proof by Berger and Wang.","confidence":0.9,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_b20a239ac0dc6a5107de6d6c91e9df51"},{"public_id":"cl_88c587d8ff8d67f0e30daaf0f0b5d39f","status":"active","text":"The generalized spectral-radius theorem states that ϱ(∑) = ρ^(∑), where ϱ(∑) is the lim sup of the m-th root of the supremum of spectral radii over m-fold products from a bounded set ∑ of complex matrices, and ρ^(∑) is the lim sup of the m-th root of the supremum of any matrix norm over those products.","confidence":0.95,"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_88c587d8ff8d67f0e30daaf0f0b5d39f"}],"concepts":[{"public_id":"co_2e048786f22e828469f970fb02bc5ce3","status":"active","name":"matrix norm","description":"Any matrix norm ‖·‖ used to define ρ^_m(∑) as the supremum over A in ∑^m.","types":["measure"],"aliases":["‖·‖"],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_2e048786f22e828469f970fb02bc5ce3"},{"public_id":"co_7224a4a3b2c76d5fd929a8a3a0c53874","status":"active","name":"spectral radius","description":"The largest absolute value of eigenvalues of a matrix, denoted ϱ(A), used to define ϱ_m(∑) as the supremum over A in ∑^m.","types":["measure"],"aliases":["ϱ"],"contributors":[{"id":32,"public_id":"7c402c1b98","public_label":"뀨 (7c402c1b98)","roles":["extraction"],"url":"https://sah.borca.ai/u/7c402c1b98"},{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["review"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_7224a4a3b2c76d5fd929a8a3a0c53874"},{"public_id":"co_806cf86a44e32a1c2a2d35fef71808d0","status":"active","name":"analytic-geometric proof","description":"A proof approach using analytic and geometric tools, presented as simpler than the original proof by 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