{"doi":"10.4153/cjm-1963-036-4","title":"On Weakly Positive Matrices","abstract":"<jats:p>A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix <jats:italic>P</jats:italic> to be positive definite is that its hermitian quadratic form</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00029539_eqn001\"/></jats:disp-formula></jats:p><jats:p>with any vector <jats:italic>v</jats:italic> ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if <jats:italic>P</jats:italic><jats:sub>1</jats:sub> and <jats:italic>P</jats:italic><jats:sub>2</jats:sub> are positive definite, the same holds of <jats:italic>a</jats:italic><jats:sub>1</jats:sub><jats:italic>P</jats:italic><jats:sub>1</jats:sub> + <jats:italic>a</jats:italic><jats:sub>2</jats:sub><jats:italic>P</jats:italic><jats:sub>2</jats:sub> if <jats:italic>a</jats:italic><jats:sub>1</jats:sub> and <jats:italic>a</jats:italic><jats:sub>2</jats:sub> are positive numbers.</jats:p>","journal":"Canadian Journal of Mathematics","year":1963,"id":625683,"datarank":0.5333022092234121,"base_score":3.5553480614894135,"endowment":3.5553480614894135,"self_citation_contribution":0.5333022092234121,"citation_network_contribution":0.0,"self_endowment_contribution":0.5333022092234121,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":34,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":1596047,"name":"Eugene P. Wigner","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"resolved":true,"title":"On Weakly Positive Matrices","abstract":"<jats:p>A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix <jats:italic>P</jats:italic> to be positive definite is that its hermitian quadratic form</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00029539_eqn001\"/></jats:disp-formula></jats:p><jats:p>with any vector <jats:italic>v</jats:italic> ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if <jats:italic>P</jats:italic><jats:sub>1</jats:sub> and <jats:italic>P</jats:italic><jats:sub>2</jats:sub> are positive definite, the same holds of <jats:italic>a</jats:italic><jats:sub>1</jats:sub><jats:italic>P</jats:italic><jats:sub>1</jats:sub> + <jats:italic>a</jats:italic><jats:sub>2</jats:sub><jats:italic>P</jats:italic><jats:sub>2</jats:sub> if <jats:italic>a</jats:italic><jats:sub>1</jats:sub> and <jats:italic>a</jats:italic><jats:sub>2</jats:sub> are positive numbers.</jats:p>","is_dataset_classified":null,"base_score":3.5553480614894135,"endowment":3.5553480614894135,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"19767382","pmcid":null,"openalex_id":"https://openalex.org/W2085581092","authors":[],"funders":[],"total_grants":0,"fwci":2.0643,"citation_percentile":0.88540541,"influential_citations":0,"citation_trend":[{"year":2012,"count":1},{"year":2016,"count":1},{"year":2017,"count":4},{"year":2018,"count":1},{"year":2021,"count":1},{"year":2024,"count":1}],"oa_status":"bronze","license":"https://www.cambridge.org/core/terms","oa_locations":[{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FE7390B19D41D25DD5FF6A6949692BEB/S0008414X00029539a.pdf/div-class-title-on-weakly-positive-matrices-div.pdf","host_type":"journal"},{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FE7390B19D41D25DD5FF6A6949692BEB/S0008414X00029539a.pdf/div-class-title-on-weakly-positive-matrices-div.pdf","host_type":"publisher"},{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00029539","host_type":"publisher"},{"url":"https://doi.org/10.4153/cjm-1963-036-4","host_type":"journal"}],"fields_of_study":["Matrix Theory and Algorithms","graph theory and CDMA systems","Advanced Mathematical Theories and Applications"],"mesh_terms":[],"keywords":["Positive-definite matrix","Mathematics","Hermitian matrix","Matrix (chemical analysis)","Pure mathematics","Quadratic equation","Definite quadratic form","Quadratic form (statistics)","Combinatorics","Eigenvalues and eigenvectors","Quadratic function","Quadratic field","Physics","Quantum mechanics","Geometry"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-08-04T07:30:42.542409Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}