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.
{"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}
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.
设A是一个n乘以n的对称矩阵。我们首先证明了如果$A$和它的伪逆是严格合的，则$A$是正半定的，推广了Han和Mangasarian的类似结果。假设A是可逆的，并且是对称的。我们在之前的一篇论文中证明了如果$A^{-1}$是非负的且$n$零对角线项，那么$A$可以是共积的(例如，这发生在Horn矩阵中)，当$A$是共积的时，它不可能是$P+ n$的形式，其中$P$是半正定的，而$n$是非负对称的。在这里，我们证明了如果$A^{-1}$是非负的且有$ N -1$ 0个对角线项和1个正对角线项，那么$A$可以是$P+N$的形式，并且我们证明了如何构造$A$。我们还证明了如果$A^{-1}$是非负的，且有1个对角线项为零，且有$n-1个对角线项为正，则$A$不可能是共生的。
{"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}
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.
{"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}
Christian Howell, Mark Kempton, Kellon Sandall, John Sinkovic
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.
{"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}
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 $mathbb{F}$.
{"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}
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.
在本文中，我们分析了具有周期系数的二维广义系统的相容性条件，并导出了一个特殊的坐标系，在该坐标系中，这些条件归结为简单的矩阵交换性条件。我们还证明，在这样一个周期性的2D描述符系统中，不同轨迹的兼容性可以很好地用所谓的规则铅笔的矩阵关系来表示，这些关系在[Benner和Byers.An algorithm for matrix pences:Theory and new algorithms.Number.Math.，103（4）:539-5732006]中介绍。然后，我们证明了这些思想可以推广到多维周期广义系统，并简要讨论了复系数矩阵和实系数矩阵情况之间的区别。
{"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}
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.
{"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}
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.
{"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}
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.
在他最近的一篇论文Molnár中，证明了如果$mathcal{A}$是一个没有$I_1, I_2$型直接和的von Neumann代数，那么从$mathcal{A}$的正定锥到保Kubo-Ando幂均值的正实数的任何函数，对于$0 neq p In(-1,1)，$必然是常数。证明了$I_1$型代数承认有非平凡的$p$幂均值保持泛函，并推测$I_2$型代数只承认有常数的$p$幂均值保持泛函。我们确认后者。在最近的另一篇关于沃瑟斯坦平均值的论文Molnár中也出现了类似的结果。我们还证明了$I_2$型代数关于Wasserstein均值的猜想。我们还给出了C^*$-代数中心性的两个条件。
{"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}
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.
{"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}
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