{"title":"Numerical Range and a Generalization of Duffin’s Overdamping Criterion","authors":"R. Hildebrand","doi":"10.1134/s0965542524700015","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The joint numerical range of tuples of matrices is a powerful tool for proving results which are useful in optimization, such as the <span>\\(\\mathcal{S}\\)</span>-lemma. Here we provide a similar proof for another result, namely the equivalence of a certain positivity criterion to Duffin’s overdamping condition involving quadratic matrix-valued polynomials. We show how the proof is generalizable to higher degrees of matrix-valued polynomials.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524700015","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The joint numerical range of tuples of matrices is a powerful tool for proving results which are useful in optimization, such as the \(\mathcal{S}\)-lemma. Here we provide a similar proof for another result, namely the equivalence of a certain positivity criterion to Duffin’s overdamping condition involving quadratic matrix-valued polynomials. We show how the proof is generalizable to higher degrees of matrix-valued polynomials.
期刊介绍：
Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
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