{"id":"W1978689577","doi":"10.1016/j.cam.2005.04.023","title":"Rank equalities for idempotent matrices with applications","year":2005,"lang":"en","type":"article","venue":"Journal of Computational and Applied Mathematics","topic":"Matrix Theory and Algorithms","field":"Computer Science","cited_by":23,"is_retracted":false,"has_abstract":false,"ca_institutions":"McGill University","funders":"","keywords":"Mathematics; Idempotence; Idempotent matrix; Rank (graph theory); Variety (cybernetics); Block (permutation group theory); Algebra over a field; Matrix (chemical analysis); Block matrix; Combinatorics; Arithmetic; Pure mathematics; Statistics; Eigenvalues and eigenvectors","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.002868185,0.001467051,0.0009532556,0.002612184,0.001402861,0.003939278,0.001360547,0.001235745,0.01616634],"category_scores_gemma":[0.02041498,0.0006673727,0.0008333125,0.002674542,0.003255405,0.007700484,0.003579295,0.003636006,0.002644719],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0006434022,"about_ca_system_score_gemma":0.0007519786,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0004190119,"about_ca_topic_score_gemma":0.0005643815,"domain_scores_codex":[0.9975603,0.0007663819,0.0001615586,0.0004157115,0.000785019,0.0003110695],"domain_scores_gemma":[0.9851746,0.009765672,0.001132677,0.001368663,0.001725726,0.0008326584],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000118899,0.00005612184,0.0002507442,0.0001059843,0.00001539144,0.0001469094,0.000265866,0.003561011,0.00170245,0.9532419,0.00545485,0.03507978],"study_design_scores_gemma":[0.0000171026,0.00003119153,0.00009869925,0.00001406225,0.000007013494,0.0001639402,0.00004845352,0.00968136,0.0008321062,0.9862241,0.002864985,0.00001714467],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.08476902,0.002688192,0.8310396,0.002385076,0.0007817514,0.00008081887,0.0004031748,0.0004818932,0.07737046],"genre_scores_gemma":[0.7606682,0.004212505,0.1762123,0.001484294,0.003246687,0.0002671558,0.0007790459,0.0005374758,0.05259241],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.01616634,"threshold_uncertainty_score":0.05408186,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01690115789551236,"score_gpt":0.2615121024575413,"score_spread":0.2446109445620289,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}