It is shown that every $2n$-by-$2n$ matrix over a field $mathbb{F}$ with determinant 1 is a product of (i) four or fewer skew-involutions ($A^2 = -I$) provided $mathbb{F} neq mathbb{Z}_3$, and (ii) eight or fewer skew-involutions if $mathbb{F} = mathbb{Z}_3$ and $n > 1$. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given.
{"title":"Products of skew-involutions","authors":"Jesus Paolo Joven, Agnes T. Paras","doi":"10.13001/ela.2023.7709","DOIUrl":"https://doi.org/10.13001/ela.2023.7709","url":null,"abstract":"It is shown that every $2n$-by-$2n$ matrix over a field $mathbb{F}$ with determinant 1 is a product of (i) four or fewer skew-involutions ($A^2 = -I$) provided $mathbb{F} neq mathbb{Z}_3$, and (ii) eight or fewer skew-involutions if $mathbb{F} = mathbb{Z}_3$ and $n > 1$. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612640","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 general linear group $mathrm{GL}(n)$ acts on the direct sum of $m$ copies of $mathrm{Mat}(n)$ by the adjoint action. The action of $mathrm{GL}(n)$ induces the action of the unitriangular subgroup $U$. We present the system of free generators of the field of $U$-invariants.
{"title":"Fields of U-invariants of matrix tuples","authors":"A. Panov","doi":"10.13001/ela.2023.7355","DOIUrl":"https://doi.org/10.13001/ela.2023.7355","url":null,"abstract":"The general linear group $mathrm{GL}(n)$ acts on the direct sum of $m$ copies of $mathrm{Mat}(n)$ by the adjoint action. The action of $mathrm{GL}(n)$ induces the action of the unitriangular subgroup $U$. We present the system of free generators of the field of $U$-invariants.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43264032","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 paper, we characterize linear bijective maps $varphi$ on the space of all $n times n$ matrices over an algebraically closed field $mathbb{F}$ having the property that the spectrum of $varphi (A)$ and $varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $varphi (A)$ and $varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.
{"title":"Linear maps that preserve parts of the spectrum on pairs of similar matrices","authors":"C. Costara","doi":"10.13001/ela.2023.7583","DOIUrl":"https://doi.org/10.13001/ela.2023.7583","url":null,"abstract":"In this paper, we characterize linear bijective maps $varphi$ on the space of all $n times n$ matrices over an algebraically closed field $mathbb{F}$ having the property that the spectrum of $varphi (A)$ and $varphi (B)$ have at least one common eigenvalue for each similar matrices $A$ and $B$. Using this result, we characterize linear bijective maps having the property that the spectrum of $varphi (A)$ and $varphi (B)$ have common elements for each matrices $A$ and $B$ having the same spectrum. As a corollary, we also characterize linear bijective maps preserving the equality of the spectrum.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48077907","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}
A. Abiad, L. de Lima, Dheer Noal Desai, Krystal Guo, L. Hogben, Jos'e Madrid
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if $G$ is a connected graph of order $n$, then $s^+(G)geq n-1$ and $s^-(G) geq n-1$. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.
{"title":"Positive and negative square energies of graphs","authors":"A. Abiad, L. de Lima, Dheer Noal Desai, Krystal Guo, L. Hogben, Jos'e Madrid","doi":"10.13001/ela.2023.7827","DOIUrl":"https://doi.org/10.13001/ela.2023.7827","url":null,"abstract":"The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if $G$ is a connected graph of order $n$, then $s^+(G)geq n-1$ and $s^-(G) geq n-1$. In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47441495","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}
Alfred Horn's conjecture on eigenvalues of sums of Hermitian matrices was proved more than 20 years ago. In this note, the problem is raised of, given an $n$-tuple $gamma$ in the solution polytope, constructing Hermitian matrices with the required spectra such that their sum has eigenvalues $gamma$.
Alfred Horn关于Hermitian矩阵和的特征值猜想是在20多年前证明的。在本文中，给出了求解多面体中的$n$-元组$gamma$，构造具有所需谱的Hermitian矩阵，使得它们的和具有特征值$gamma$的问题。
{"title":"The inverse Horn problem","authors":"J. Queiró, A. P. Santana","doi":"10.13001/ela.2023.7539","DOIUrl":"https://doi.org/10.13001/ela.2023.7539","url":null,"abstract":"Alfred Horn's conjecture on eigenvalues of sums of Hermitian matrices was proved more than 20 years ago. In this note, the problem is raised of, given an $n$-tuple $gamma$ in the solution polytope, constructing Hermitian matrices with the required spectra such that their sum has eigenvalues $gamma$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46228169","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 paper, we prove an inequality involving the permanent of a positive semidefinite matrix and its leading submatrices. We obtain a result in the similar spirit of Bapat-Sunder per-max conjecture.
{"title":"A permanent inequality for positive semidefinite matrices","authors":"Vehbi E. Paksoy","doi":"10.13001/ela.2023.7701","DOIUrl":"https://doi.org/10.13001/ela.2023.7701","url":null,"abstract":"In this paper, we prove an inequality involving the permanent of a positive semidefinite matrix and its leading submatrices. We obtain a result in the similar spirit of Bapat-Sunder per-max conjecture.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46162923","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 $(k+1)$-core of a graph $G$, denoted by $C_{k+1}(G)$, is the subgraph obtained by repeatedly removing any vertex of degree less than or equal to $k$. $C_{k+1}(G)$ is the unique induced subgraph of minimum degree larger than $k$ with a maximum number of vertices. For $1leq kleq mleq n$, we denote $R_{n, k, m}=K_kvee(K_{m-k}cup {I_{n-m}})$. In this paper, we prove that $R_{n, k, m}$ obtains the maximum spectral radius and signless Laplacian spectral radius among all $n$-vertex graphs whose $(k+1)$-core has at most $m$ vertices. Our result extends a recent theorem proved by Nikiforov [Electron. J. Linear Algebra, 27:250--257, 2014]. Moreover, we also present the bipartite version of our result.
{"title":"The maximum spectral radius of graphs with a large core","authors":"Xiaocong He, Lihua Feng, D. Stevanović","doi":"10.13001/ela.2023.7283","DOIUrl":"https://doi.org/10.13001/ela.2023.7283","url":null,"abstract":"The $(k+1)$-core of a graph $G$, denoted by $C_{k+1}(G)$, is the subgraph obtained by repeatedly removing any vertex of degree less than or equal to $k$. $C_{k+1}(G)$ is the unique induced subgraph of minimum degree larger than $k$ with a maximum number of vertices. For $1leq kleq mleq n$, we denote $R_{n, k, m}=K_kvee(K_{m-k}cup {I_{n-m}})$. In this paper, we prove that $R_{n, k, m}$ obtains the maximum spectral radius and signless Laplacian spectral radius among all $n$-vertex graphs whose $(k+1)$-core has at most $m$ vertices. Our result extends a recent theorem proved by Nikiforov [Electron. J. Linear Algebra, 27:250--257, 2014]. Moreover, we also present the bipartite version of our result.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47160286","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 $G$ be a strongly connected digraph with vertex set ${v_1, v_2, dots, v_n}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of the distance matrix were proposed by Yan and Yeh (Adv. Appl. Math.), and Bapat et al. (Linear Algebra Appl.) independently, one is the $q$-distance matrix, and the other is the exponential distance matrix. Given a nonzero indeterminate $q$, the $q$-distance matrix $mathscr{D}_G=(mathscr{D}_{ij})_{ntimes n}$ of $G$ is defined as[mathscr{D}_{ij}=left{begin{array}{cl}1+q+dots+q^{D_{ij}-1}&text{if $ine j$},0&text{otherwise}.end{array}right.]In particular, when $q = 1$, it would be reduced to the distance matrix of $G$. The exponential distance matrix $mathscr{F}_G=(mathscr{F}_{ij})_{ntimes n}$ of $G$ is defined as[mathscr{F}_{ij}= q^{D_{ij}}.] In $1977$, Graham et al. (J. Graph Theory) established a classical formula connecting the determinants and cofactor sums of the distance matrices of strongly connected digraphs in terms of their blocks, which plays a powerful role in the subsequent researches on the determinants of distance matrices. Sivasubramanian (Electron. J. Combin.) and Li et al. (Discuss. Math. Graph Theory) independently extended it from the distance matrix to the $q$-distance matrix. In this note, three formulae of such types for the exponential distance matrices of strongly connected digraphs will be presented.
{"title":"The Graham-Hoffman-Hosoya-type theorems for the exponential distance matrix","authors":"Z. Du, Rundan Xing","doi":"10.13001/ela.2023.7449","DOIUrl":"https://doi.org/10.13001/ela.2023.7449","url":null,"abstract":"Let $G$ be a strongly connected digraph with vertex set ${v_1, v_2, dots, v_n}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of the distance matrix were proposed by Yan and Yeh (Adv. Appl. Math.), and Bapat et al. (Linear Algebra Appl.) independently, one is the $q$-distance matrix, and the other is the exponential distance matrix. Given a nonzero indeterminate $q$, the $q$-distance matrix $mathscr{D}_G=(mathscr{D}_{ij})_{ntimes n}$ of $G$ is defined as[mathscr{D}_{ij}=left{begin{array}{cl}1+q+dots+q^{D_{ij}-1}&text{if $ine j$},0&text{otherwise}.end{array}right.]In particular, when $q = 1$, it would be reduced to the distance matrix of $G$. The exponential distance matrix $mathscr{F}_G=(mathscr{F}_{ij})_{ntimes n}$ of $G$ is defined as[mathscr{F}_{ij}= q^{D_{ij}}.] In $1977$, Graham et al. (J. Graph Theory) established a classical formula connecting the determinants and cofactor sums of the distance matrices of strongly connected digraphs in terms of their blocks, which plays a powerful role in the subsequent researches on the determinants of distance matrices. Sivasubramanian (Electron. J. Combin.) and Li et al. (Discuss. Math. Graph Theory) independently extended it from the distance matrix to the $q$-distance matrix. In this note, three formulae of such types for the exponential distance matrices of strongly connected digraphs will be presented.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41614980","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 number of flat portions on the boundary of the numerical range of $5 times 5$ companion matrices, both unitarily reducible and unitarily irreducible cases, is examined. The complete characterization on the number of flat portions of a $5 times 5$ unitarily reducible companion matrix is given. Also under some suitable conditions, it is shown that a unitarily irreducible $5 times 5$ companion matrix cannot have four flat portions on the boundary of its numerical range. This gives a partial affirmative answer to the conjecture given in [3] for $n = 5$. Numerical examples are provided to illustrate the results.
{"title":"Flat portions on the boundary of the numerical range of a 5 × 5 companion matrix","authors":"Swastika Saha Mondal, Sarita Ojha, R. Birbonshi","doi":"10.13001/ela.2023.7209","DOIUrl":"https://doi.org/10.13001/ela.2023.7209","url":null,"abstract":"The number of flat portions on the boundary of the numerical range of $5 times 5$ companion matrices, both unitarily reducible and unitarily irreducible cases, is examined. The complete characterization on the number of flat portions of a $5 times 5$ unitarily reducible companion matrix is given. Also under some suitable conditions, it is shown that a unitarily irreducible $5 times 5$ companion matrix cannot have four flat portions on the boundary of its numerical range. This gives a partial affirmative answer to the conjecture given in [3] for $n = 5$. Numerical examples are provided to illustrate the results.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49036387","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 square matrix $A$, it is proved that minimal rank weak Drazin inverses (Campbell and Meyer, 1978) of $A$ coincide with outer inverses of $A$ with range $mathcal{R}(A^{k})$, where $k$ is the index of $A$. It is shown that the minimal rank weak Drazin inverse behaves very much like the Drazin inverse, and many generalized inverses such as the core-EP inverse and the DMP inverse are its special cases.
对于任意方阵$A$，证明了$A$的最小秩弱Drazin逆(Campbell and Meyer, 1978)与$A$的外逆在$mathcal{R}(A^{k})$范围内一致，其中$k$是$A$的索引。证明了极小阶弱Drazin逆的性质与Drazin逆非常相似，许多广义逆如core-EP逆和DMP逆都是它的特例。
{"title":"Minimal rank weak Drazin inverses: a class of outer inverses with prescribed range","authors":"Cang Wu, Jianlong Chen","doi":"10.13001/ela.2023.7359","DOIUrl":"https://doi.org/10.13001/ela.2023.7359","url":null,"abstract":"For any square matrix $A$, it is proved that minimal rank weak Drazin inverses (Campbell and Meyer, 1978) of $A$ coincide with outer inverses of $A$ with range $mathcal{R}(A^{k})$, where $k$ is the index of $A$. It is shown that the minimal rank weak Drazin inverse behaves very much like the Drazin inverse, and many generalized inverses such as the core-EP inverse and the DMP inverse are its special cases.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47028513","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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