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Diagonal-Schur complements of Nekrasov matrices Nekrasov矩阵的对角-舒尔补
4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-10-24 DOI: 10.13001/ela.2023.7941
Shiyun Wang, Qi Li, Xu Sun, Zhenhua Lyu
{"title":"Diagonal-Schur complements of Nekrasov matrices","authors":"Shiyun Wang, Qi Li, Xu Sun, Zhenhua Lyu","doi":"10.13001/ela.2023.7941","DOIUrl":"https://doi.org/10.13001/ela.2023.7941","url":null,"abstract":"The Schur and diagonal-Schur complements are important tools in many fields. It was revealed that the diagonal-Schur complements of Nekrasov matrices with respect to the index set ${1}$ are Nekrasov matrices by Cvetkovic and Nedovic [Appl. Math. Comput., 208:225-230, 2009]. In this paper, we prove that the diagonal-Schur complements of Nekrasov matrices with respect to any index set are Nekrasov matrices. Similar results hold for $Sigma$-Nekrasov matrices. We also present some results on Nekrasov diagonally dominant degrees. Numerical examples are given to verify the correctness of the results.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135268332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The inverse of a symmetric nonnegative matrix can be copositive 对称非负矩阵的逆可以是共积的
4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-09-27 DOI: 10.13001/ela.2023.7927
Robert Reams
{"title":"The inverse of a symmetric nonnegative matrix can be copositive","authors":"Robert Reams","doi":"10.13001/ela.2023.7927","DOIUrl":"https://doi.org/10.13001/ela.2023.7927","url":null,"abstract":"Let $A$ be an $ntimes n$ symmetric matrix. We first show that if $A$ and its pseudoinverse are strictly copositive, then $A$ is positive semidefinite, which extends a similar result of Han and Mangasarian. Suppose $A$ is invertible, as well as being symmetric. We showed in an earlier paper that if $A^{-1}$ is nonnegative with $n$ zero diagonal entries, then $A$ can be copositive (for instance, this happens with the Horn matrix), and when $A$ is copositive, it cannot be of form $P+N$, where $P$ is positive semidefinite and $N$ is nonnegative and symmetric. Here, we show that if $A^{-1}$ is nonnegative with $n-1$ zero diagonal entries and one positive diagonal entry, then $A$ can be of the form $P+N$, and we show how to construct $A$. We also show that if $A^{-1}$ is nonnegative with one zero diagonal entry and $n-1$ positive diagonal entries, then $A$ cannot be copositive.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135535460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On condition numbers of quaternion matrix inverse and quaternion linear systems with multiple right-hand sides 四元数矩阵逆和多右手边四元数线性系统的条件数
4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-09-26 DOI: 10.13001/ela.2023.7799
Qiaohua Liu, Shan Wang, Fengxia Zhang
{"title":"On condition numbers of quaternion matrix inverse and quaternion linear systems with multiple right-hand sides","authors":"Qiaohua Liu, Shan Wang, Fengxia Zhang","doi":"10.13001/ela.2023.7799","DOIUrl":"https://doi.org/10.13001/ela.2023.7799","url":null,"abstract":"This paper is devoted to the condition numbers of quaternion linear system with multiple right-hand sides and the associated condition numbers of the quaternion matrix inverse as well. The explicit expressions of the unstructured and structured normwise, mixed, and componentwise condition numbers for the system are given. To reduce the computational cost of the condition numbers,compact and tight upper bounds for these condition numbers are proposed. For general sparse and badly scaled problems, numerical examples show that mixed and componentwise condition numbers are preferred than the normwise condition number for estimating the forward error of the solution, and structured condition numbers are tighter than the unstructured ones for some specific structured problems.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134961122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unicyclic graphs and the inertia of the squared distance matrix 单环图和距离平方矩阵的惯性
4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-09-11 DOI: 10.13001/ela.2023.7543
Christian Howell, Mark Kempton, Kellon Sandall, John Sinkovic
{"title":"Unicyclic graphs and the inertia of the squared distance matrix","authors":"Christian Howell, Mark Kempton, Kellon Sandall, John Sinkovic","doi":"10.13001/ela.2023.7543","DOIUrl":"https://doi.org/10.13001/ela.2023.7543","url":null,"abstract":"A result of Bapat and Sivasubramanian gives the inertia of the squared distance matrix of a tree. We develop general tools on how pendant vertices and vertices of degree 2 affect the inertia of the squared distance matrix and use these to give an alternative proof of this result. We further use these tools to extend this result to certain families of unicyclic graphs, and we explore how far these results can be extended.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135982868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence 矩阵分解为可逆矩阵和固定幂零矩阵的和
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-08-24 DOI: 10.13001/ela.2023.7851
P. Danchev, E. García, M. Gómez Lozano
{"title":"Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence","authors":"P. Danchev, E. García, M. Gómez Lozano","doi":"10.13001/ela.2023.7851","DOIUrl":"https://doi.org/10.13001/ela.2023.7851","url":null,"abstract":"For any $nge 2$ and fixed $kge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $mathbb{M}_n(mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $mathbb{F}$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43378041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Periodic two-dimensional descriptor systems 周期二维广义系统
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-08-24 DOI: 10.13001/ela.2023.7989
P. Benner, P. Van Dooren
{"title":"Periodic two-dimensional descriptor systems","authors":"P. Benner, P. Van Dooren","doi":"10.13001/ela.2023.7989","DOIUrl":"https://doi.org/10.13001/ela.2023.7989","url":null,"abstract":"In this note, we analyze the compatibility conditions of 2D descriptor systems with periodic coefficients and we derive a special coordinate system in which these conditions reduce to simple matrix commutativity conditions. We also show that the compatibility of the different trajectories in such a periodic 2D descriptor system can elegantly be formulated in terms of so-called matrix relations of regular pencils, which were introduced in [Benner and Byers. An arithmetic for matrix pencils: Theory and new algorithms. Numer. Math., 103(4):539-573, 2006]. We then show that these ideas can be extended to multidimensional periodic descriptor systems and briefly discuss the difference between the case of complex and real coefficient matrices.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43506986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Pareto H-eigenvalues of nonnegative tensors and uniform hypergraphs 非负张量和一致超图的Pareto h特征值
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-08-02 DOI: 10.13001/ela.2023.7839
Lu Zheng, Bo Zhou
{"title":"Pareto H-eigenvalues of nonnegative tensors and uniform hypergraphs","authors":"Lu Zheng, Bo Zhou","doi":"10.13001/ela.2023.7839","DOIUrl":"https://doi.org/10.13001/ela.2023.7839","url":null,"abstract":"The Pareto H-eigenvalues of nonnegative tensors and (adjacency tensors of) uniform hypergraphs are studied. Particularly, it is shown that the Pareto H-eigenvalues of a nonnegative tensor are just the spectral radii of its weakly irreducible principal subtensors, and those hypergraphs that minimize or maximize the second largest Pareto H-eigenvalue over several well-known classes of uniform hypergraphs are determined.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45597720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs 有向图的拉普拉斯特征值的k个最大绝对值之和
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-07-20 DOI: 10.13001/ela.2023.7503
Xiuwen Yang, Xiaogang Liu, Ligong Wang
{"title":"On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs","authors":"Xiuwen Yang, Xiaogang Liu, Ligong Wang","doi":"10.13001/ela.2023.7503","DOIUrl":"https://doi.org/10.13001/ela.2023.7503","url":null,"abstract":"Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ be the sum of the $k$ largest absolute values of Laplacian eigenvalues of $G$. Let $C_n^+$ be a digraph with $n+1$ vertices obtained from the directed cycle $C_n$ by attaching a pendant arc whose tail is on $C_n$. A digraph is $mathbb{C}_n^+$-free if it contains no $C_{ell}^+$ as a subdigraph for any $2leq ell leq n-1$. In this paper, we present lower bounds of $S_n(G)$ of digraphs of order $n$. We provide the exact values of $S_k(G)$ of directed cycles and $mathbb{C}_n^+$-free unicyclic digraphs. Moreover, we obtain upper bounds of $S_k(G)$ of $mathbb{C}_n^+$-free digraphs which have vertex-disjoint directed cycles.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44092532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Preservers of the p-power and the Wasserstein means on 2x2 matrices 2x2矩阵上p-幂的保持器和Wasserstein均值
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-07-13 DOI: 10.13001/ela.2023.7679
R. Simon, Dániel Virosztek
{"title":"Preservers of the p-power and the Wasserstein means on 2x2 matrices","authors":"R. Simon, Dániel Virosztek","doi":"10.13001/ela.2023.7679","DOIUrl":"https://doi.org/10.13001/ela.2023.7679","url":null,"abstract":"In one of his recent papers, Molnár showed that if $mathcal{A}$ is a von Neumann algebra without $I_1, I_2$-type direct summands, then any function from the positive definite cone of $mathcal{A}$ to the positive real numbers preserving the Kubo-Ando power mean, for some $0 neq p in (-1,1),$ is necessarily constant. It was shown in that paper that $I_1$-type algebras admit nontrivial $p$-power mean preserving functionals, and it was conjectured that $I_2$-type algebras admit only constant $p$-power mean preserving functionals. We confirm the latter. A similar result occurred in another recent paper of Molnár concerning the Wasserstein mean. We prove the conjecture for $I_2$-type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in $C^*$-algebras.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48364989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Reverse order law and forward order law for the (b, c)-inverse (b, c)-逆的逆序律和正序律
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2023-07-13 DOI: 10.13001/ela.2023.7807
Jelena Višnjić, Ivana Stanisev, Y. Ke
{"title":"Reverse order law and forward order law for the (b, c)-inverse","authors":"Jelena Višnjić, Ivana Stanisev, Y. Ke","doi":"10.13001/ela.2023.7807","DOIUrl":"https://doi.org/10.13001/ela.2023.7807","url":null,"abstract":"The reverse order law and the forward order law have been studied for various types of generalized inverses. The $(b,c)$-inverse is a generalization of some well known generalized inverses, such as the Moore-Penrose inverse, the Drazin inverse, the core inverse, etc. In this paper, the reverse order law for the $(b,c)$-inverse, in a unital ring, is investigated and an equivalent condition for this law to hold for the $(b,c)$-inverse is derived. Also, some known results on this topic are generalized. Furthermore, the forward order law for the $(b,c)$-inverse in a ring with a unity is introduced, for different choices of $b$ and $c$. Moreover, as corollaries of obtained results, equivalent conditions for the reverse order law and the forward order law for the inverse along an element are derived.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42120640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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