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Electronic Journal of Linear Algebra最新文献

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A note on the boundary of the Birkhoff-James ε-orthogonality sets 关于Birkhoff-James ε-正交集边界的一个注记
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2022-01-12 DOI: 10.13001/ela.2022.6561
Georgios Katsouleas, Vasiliki Panagakou, P. Psarrakos
{"title":"A note on the boundary of the Birkhoff-James ε-orthogonality sets","authors":"Georgios Katsouleas, Vasiliki Panagakou, P. Psarrakos","doi":"10.13001/ela.2022.6561","DOIUrl":"https://doi.org/10.13001/ela.2022.6561","url":null,"abstract":"The Birkhoff-James $varepsilon$-sets of vectors and vector-valued polynomials (in one complex variable) have recently been introduced as natural generalizations of the standard numerical range of (square) matrices or operators and matrix or operator polynomials, respectively. Corners on the boundary curves of these sets are of particular interest, not least because of their importance in visualizing these sets. In this paper, we provide a characterization for the corners of the Birkhoff-James $varepsilon$-sets of vectors and vector-valued polynomials, completing and expanding upon previous exploration of the geometric propertiesof these sets. We also propose a randomized algorithm for approximating their boundaries.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47004641","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
Real dimension of the Lie algebra of S-skew-Hermitian matrices S-偏序矩阵李代数的实维数
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2022-01-07 DOI: 10.13001/ela.2022.5443
Jonathan Caalim, Yuuji Tanaka
{"title":"Real dimension of the Lie algebra of S-skew-Hermitian matrices","authors":"Jonathan Caalim, Yuuji Tanaka","doi":"10.13001/ela.2022.5443","DOIUrl":"https://doi.org/10.13001/ela.2022.5443","url":null,"abstract":"Let $M_n(mathbb{C})$ be the set of $ntimes n$ matrices over the complex numbers. Let $S in M_n(mathbb{C})$. A matrix $Ain M_n(mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600907","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
Compatibility and companions for Leonard pairs Leonard配对的兼容性和同伴
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-26 DOI: 10.13001/ela.2022.6861
K. Nomura, Paul M. Terwilliger
{"title":"Compatibility and companions for Leonard pairs","authors":"K. Nomura, Paul M. Terwilliger","doi":"10.13001/ela.2022.6861","DOIUrl":"https://doi.org/10.13001/ela.2022.6861","url":null,"abstract":"In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $mathbb{F}$ denote a field, and let $V$ denote a vector space over $mathbb{F}$ with finite positive dimension.A Leonard pair on $V$ is an ordered pair of diagonalizable $mathbb{F}$-linear maps $A : V to V$ and $A^* : V to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $mathbb{F}$-linear map $K: V to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$,define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$.For each solution $B, B^*$, we describe the corresponding companion $K = A-B$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44289520","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
The algebra generated by nilpotent elements in a matrix centralizer 矩阵中心化子中幂零元生成的代数
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-23 DOI: 10.13001/ela.2022.6503
Ralph John de la Cruz, Eloise Misa
{"title":"The algebra generated by nilpotent elements in a matrix centralizer","authors":"Ralph John de la Cruz, Eloise Misa","doi":"10.13001/ela.2022.6503","DOIUrl":"https://doi.org/10.13001/ela.2022.6503","url":null,"abstract":"For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41448142","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: 2
A note on commuting additive maps on rank k symmetric matrices 关于k阶对称矩阵上可交换加性映射的注释
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-22 DOI: 10.13001/ela.2021.6349
W. L. Chooi, Yean Nee Tan
{"title":"A note on commuting additive maps on rank k symmetric matrices","authors":"W. L. Chooi, Yean Nee Tan","doi":"10.13001/ela.2021.6349","DOIUrl":"https://doi.org/10.13001/ela.2021.6349","url":null,"abstract":"Let $ngeq 2$ and $1<kleq n$ be integers. Let $S_n(mathbb{F})$ be the linear space of $ntimes n$ symmetric matrices over a field $mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $psi:S_n(mathbb{F})rightarrow S_n(mathbb{F})$ satisfies $psi(A)A=Apsi(A)$ for all rank $k$ matrices $Ain S_n(mathbb{F})$ if and only if there exists a scalar $lambdain mathbb{F}$ and an additive map $mu:S_n(mathbb{F})rightarrow mathbb{F}$ such that[psi(A)=lambda A+mu(A)I_n,]for all $Ain S_n(mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $mathbb{F}$ of characteristic not two are included.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43116804","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
Inertia Sets of Semicliqued Graphs 半液化图的惯性集
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-22 DOI: 10.13001/ela.2021.4933
A. Yielding, Taylor Hunt, Joel Jacobs, Jazmine Juarez, Taylor Rhoton, Heath Sell
{"title":"Inertia Sets of Semicliqued Graphs","authors":"A. Yielding, Taylor Hunt, Joel Jacobs, Jazmine Juarez, Taylor Rhoton, Heath Sell","doi":"10.13001/ela.2021.4933","DOIUrl":"https://doi.org/10.13001/ela.2021.4933","url":null,"abstract":"In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $widetilde{K}G$. We establish the minimum rank and inertia sets of some $widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45337489","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
Expressions and characterizations for the Moore-Penrose inverse of operators and matrices 算子和矩阵的Moore-Penrose逆的表达式和表征
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-20 DOI: 10.13001/ela.2023.7315
P. Morillas
{"title":"Expressions and characterizations for the Moore-Penrose inverse of operators and matrices","authors":"P. Morillas","doi":"10.13001/ela.2023.7315","DOIUrl":"https://doi.org/10.13001/ela.2023.7315","url":null,"abstract":"Under certain conditions, we prove that the Moore-Penrose inverse of a sum of operators is the sum of the Moore-Penrose inverses. From this, we derive expressions and characterizations for the Moore-Penrose inverse of an operator that are useful for its computation. We give formulations of them for finite matrices and study the Moore-Penrose inverse of circulant matrices and of distance matrices of certain graphs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43032973","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
On $m$-th roots of nilpotent matrices 关于幂零矩阵的$m$-根
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-14 DOI: 10.13001/ela.2021.6331
Semra Ozturk
{"title":"On $m$-th roots of nilpotent matrices","authors":"Semra Ozturk","doi":"10.13001/ela.2021.6331","DOIUrl":"https://doi.org/10.13001/ela.2021.6331","url":null,"abstract":"A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42388324","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 Gerv{s}gorin disks of distance matrices of graphs 图的距离矩阵的Gerv{s}gorin盘
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-08 DOI: 10.13001/ela.2021.6489
M. Aouchiche, B. Rather, Issmail El Hallaoui
{"title":"On the Gerv{s}gorin disks of distance matrices of graphs","authors":"M. Aouchiche, B. Rather, Issmail El Hallaoui","doi":"10.13001/ela.2021.6489","DOIUrl":"https://doi.org/10.13001/ela.2021.6489","url":null,"abstract":"For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Gerv{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46275494","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
Families of graphs with twin pendent paths and the Braess edge 具有双悬垂路径和Braess边的图族
IF 0.7 4区 数学
Electronic Journal of Linear Algebra Pub Date : 2021-12-07 DOI: 10.13001/ela.2022.5913
Sooyeon Kim
{"title":"Families of graphs with twin pendent paths and the Braess edge","authors":"Sooyeon Kim","doi":"10.13001/ela.2022.5913","DOIUrl":"https://doi.org/10.13001/ela.2022.5913","url":null,"abstract":"In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to Kemeny's constant: in particular, we examine graphs with a cut-vertex at which at least two branches are paths, regarding whether the insertion of a particular edge into a graph results in an increase of Kemeny's constant. We provide several tools for identifying such an edge in a family of graphs and for analysing asymptotic behaviour of the family regarding the tendency to have that edge; and classes of particular graphs are given as examples. Furthermore, asymptotic behaviours of families of trees are described.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42491726","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: 4
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