{"id":"https://openalex.org/W2061199615","doi":"https://doi.org/10.1137/060658126","title":"On Pseudospectra and Power Growth","display_name":"On Pseudospectra and Power Growth","publication_year":2007,"publication_date":"2007-01-01","ids":{"openalex":"https://openalex.org/W2061199615","doi":"https://doi.org/10.1137/060658126","mag":"2061199615"},"language":"en","primary_location":{"id":"doi:10.1137/060658126","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060658126","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5033632050","display_name":"Thomas Ransford","orcid":"https://orcid.org/0000-0003-0693-6937"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thomas Ransford","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5033632050"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3736,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.64361638,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"29","issue":"3","first_page":"699","last_page":"711"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9854999780654907,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9686999917030334,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8617482781410217},{"id":"https://openalex.org/keywords/resolvent","display_name":"Resolvent","score":0.8188764452934265},{"id":"https://openalex.org/keywords/diagonalizable-matrix","display_name":"Diagonalizable matrix","score":0.7620737552642822},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.6443090438842773},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.5875058770179749},{"id":"https://openalex.org/keywords/complex-matrix","display_name":"Complex matrix","score":0.4981651306152344},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4942305386066437},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4645533263683319},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4634881317615509},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.46088218688964844},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4594252109527588},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4420890808105469},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.3553343713283539},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.329404354095459},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.17345276474952698},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.062019675970077515}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8617482781410217},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.8188764452934265},{"id":"https://openalex.org/C30072841","wikidata":"https://www.wikidata.org/wiki/Q1767080","display_name":"Diagonalizable matrix","level":4,"score":0.7620737552642822},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.6443090438842773},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.5875058770179749},{"id":"https://openalex.org/C2988629283","wikidata":"https://www.wikidata.org/wiki/Q44337","display_name":"Complex matrix","level":2,"score":0.4981651306152344},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4942305386066437},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4645533263683319},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4634881317615509},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46088218688964844},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4594252109527588},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4420890808105469},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.3553343713283539},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.329404354095459},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.17345276474952698},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.062019675970077515},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/060658126","is_oa":false,"landing_page_url":"https://doi.org/10.1137/060658126","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:uuid:7781c019-d4b8-45bf-b3c7-0340a51c28f7","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"http://eprints.maths.ox.ac.uk/1119","raw_type":"Technical Report"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.992.7983","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.992.7983","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.maths.ox.ac.uk/1119/1/NA-06-04.pdf","raw_type":"text"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:7781c019-d4b8-45bf-b3c7-0340a51c28f7","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:7781c019-d4b8-45bf-b3c7-0340a51c28f7","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.maths.ox.ac.uk/1119","raw_type":"Report"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.44999998807907104,"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W149522324","https://openalex.org/W175988912","https://openalex.org/W409772452","https://openalex.org/W1486267208","https://openalex.org/W1511607736"],"related_works":["https://openalex.org/W4366831400","https://openalex.org/W2158044763","https://openalex.org/W4298234822","https://openalex.org/W2782047334","https://openalex.org/W3017414697","https://openalex.org/W2006601391","https://openalex.org/W2962791795","https://openalex.org/W3013272832","https://openalex.org/W4287823279","https://openalex.org/W3144502749"],"abstract_inverted_index":{"The":[0,120],"celebrated":[1],"Kreiss":[2],"matrix":[3,18],"theorem":[4],"is":[5,72],"one":[6],"of":[7,13,16,26,29,43],"several":[8],"results":[9],"relating":[10],"the":[11,14,23,27,30,37,41,44,57,86,100,116],"norms":[12,42,113],"powers":[15],"a":[17,133],"to":[19,33,56,122,128],"its":[20],"pseudospectra":[21,38],"(i.e.,":[22],"level":[24],"curves":[25],"norm":[28,60,101],"resolvent).":[31],"But":[32],"what":[34],"extent":[35],"do":[36],"actually":[39],"determine":[40],"powers?":[45],"Specifically,":[46],"let":[47],"$A,B$":[48,90],"be":[49,129],"square":[50],"matrices":[51],"such":[52],"that,":[53],"with":[54],"respect":[55],"usual":[58,117],"operator":[59,118],"$\\|\\cdot\\|$,":[61,102],"we":[62,111],"have":[63],"$\\|(zI-A)^{-1}\\|=\\|(zI-B)^{-1}\\|$":[64],"$(z\\in{\\mathbb":[65],"C}).$":[66],"(Call":[67],"this":[68],"$(*)$.)":[69],"Then":[70],"it":[71],"known":[73],"that":[74],"$1/2\\le\\|A\\|/\\|B\\|\\le":[75],"2$.":[76],"Are":[77],"there":[78],"similar":[79],"bounds":[80],"for":[81,83,99,105],"$\\|A^n\\|/\\|B^n\\|$":[82],"$n\\ge2$?":[84],"Does":[85],"answer":[87],"change":[88],"if":[89,94,110],"are":[91],"diagonalizable?":[92],"What":[93,109],"$(*)$":[95],"holds,":[96],"not":[97],"just":[98],"but":[103],"also":[104],"higher-order":[106],"singular":[107],"values?":[108],"use":[112],"other":[114],"than":[115],"norm?":[119],"answers":[121],"all":[123],"these":[124],"questions":[125],"turn":[126],"out":[127],"negative,":[130],"and":[131],"in":[132],"rather":[134],"strong":[135],"sense.":[136]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}