Pub Date : 2019-07-10DOI: 10.4236/ALAMT.2019.93003
Wasim Audeh
A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.
， n, f,g为[0，∞)上的非负连续函数，满足关系f(t)g(t) = t (t∈[0，∞))，则对于所有r≥1。， n和f,g是[0，∞)上的非负连续函数，满足关系f(t)g(t) = t (t∈[0，∞))，则其中。此外，我们还给出了许多比最近证明的相关不等式更尖锐的数值半径不等式，并给出了一些应用。
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Pub Date : 2019-06-30DOI: 10.4236/ALAMT.2019.92002
A. Dax
Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincare are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.
Ky Fan迹定理和柯西和庞加莱的交错定理是表征厄米矩阵的重要观察结果。在这篇笔记中，我们引入一类新的不等式来扩展这些定理。新的不等式是通过用它们的模替换特征值和对角线项而得到的。这种修正产生了有效的边界不等式，它在更大范围的矩阵上有效。
{"title":"Bounding Inequalities for Eigenvalues of Principal Submatrices","authors":"A. Dax","doi":"10.4236/ALAMT.2019.92002","DOIUrl":"https://doi.org/10.4236/ALAMT.2019.92002","url":null,"abstract":"Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincare are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81528072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-29DOI: 10.4236/ALAMT.2019.91001
Jean-Francois Niglio
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on by using the rotation group [3] [4]. It will be proved that the group acts on elements of in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.
{"title":"A Follow-Up on Projection Theory: Theorems and Group Action","authors":"Jean-Francois Niglio","doi":"10.4236/ALAMT.2019.91001","DOIUrl":"https://doi.org/10.4236/ALAMT.2019.91001","url":null,"abstract":"In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on by using the rotation group [3] [4]. It will be proved that the group acts on elements of in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85502910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4236/alamt.2019.94006
H. Liu, Honglian Zhang
The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.
{"title":"A Minimality of the Rational Canonical Form","authors":"H. Liu, Honglian Zhang","doi":"10.4236/alamt.2019.94006","DOIUrl":"https://doi.org/10.4236/alamt.2019.94006","url":null,"abstract":"The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80432754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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