Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, have been extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.
{"title":"Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem","authors":"N. T. Son, T. Stykel","doi":"10.13001/ela.2022.7351","DOIUrl":"https://doi.org/10.13001/ela.2022.7351","url":null,"abstract":"Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, have been extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46701831","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 $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher et al. proved that for any tree $T$ of order $n$, $S_2(T) leq n+2-frac{2}{n}$. Guan et al. determined the tree with maximum $S_2(T)$ among all trees of order $n$. In this paper, we characterize the trees with $S_2(T) geq n+1$ among all trees of order $n$ except some trees. Moreover, among all trees of order $n$, we also determine the first $lfloorfrac{n-2}{2}rfloor$ trees according to their $S_2(T)$. This extends the result of Guan et al.
{"title":"Trees with maximum sum of the two largest Laplacian eigenvalues","authors":"Yirong Zheng, Jianxi Li, Sarula Chang","doi":"10.13001/ela.2022.7065","DOIUrl":"https://doi.org/10.13001/ela.2022.7065","url":null,"abstract":"Let $T$ be a tree of order $n$ and $S_2(T)$ be the sum of the two largest Laplacian eigenvalues of $T$. Fritscher et al. proved that for any tree $T$ of order $n$, $S_2(T) leq n+2-frac{2}{n}$. Guan et al. determined the tree with maximum $S_2(T)$ among all trees of order $n$. In this paper, we characterize the trees with $S_2(T) geq n+1$ among all trees of order $n$ except some trees. Moreover, among all trees of order $n$, we also determine the first $lfloorfrac{n-2}{2}rfloor$ trees according to their $S_2(T)$. This extends the result of Guan et al.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45727456","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}
We obtain some inequalities involving positive linear maps on matrix algebra. The special cases provide bounds for the spreads of normal matrices.
在矩阵代数上得到了一些涉及正线性映射的不等式。特殊情况为正常矩阵的扩展提供了界。
{"title":"Positive linear maps and spreads of normal matrices","authors":"Rajesh Sharma, Manish Pal","doi":"10.13001/ela.2022.7009","DOIUrl":"https://doi.org/10.13001/ela.2022.7009","url":null,"abstract":"We obtain some inequalities involving positive linear maps on matrix algebra. The special cases provide bounds for the spreads of normal matrices.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43315655","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 give a sufficient and necessary condition for the subdirect sum of a Nekrasov matrix and a strictly diagonally dominant matrix being still a Nekrasov matrix. Adopting this sufficient and necessary condition, we present several simple sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the same class. Examples are reported to illustrate the theoretical results.
{"title":"K-subdirect sums of Nekrasov matrices","authors":"Zhenhua Lyu, Xueru Wang, Lishu Wen","doi":"10.13001/ela.2022.6951","DOIUrl":"https://doi.org/10.13001/ela.2022.6951","url":null,"abstract":"In this paper, we give a sufficient and necessary condition for the subdirect sum of a Nekrasov matrix and a strictly diagonally dominant matrix being still a Nekrasov matrix. Adopting this sufficient and necessary condition, we present several simple sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the same class. Examples are reported to illustrate the theoretical results.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45950983","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}
Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $nleq 4$ but fail starting with $n=5$.
{"title":"On low-dimensional partial isometries","authors":"Qixiao He, I. Spitkovsky, I. Suleiman","doi":"10.13001/ela.2023.7405","DOIUrl":"https://doi.org/10.13001/ela.2023.7405","url":null,"abstract":"Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $nleq 4$ but fail starting with $n=5$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49293101","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}
Classes of an equivalence relation on a module $V$ over a supertropical semiring, called rays, carry the underlying structure of 'supertropical trigonometry' and thereby a version of convex geometry which is compatible with quasilinearity. In this theory, the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio, which gives rise to special characteristic functions, called CS-functions. These functions partite the ray space $mathrm{Ray}(V)$ into convex sets and establish the main tool for analyzing varieties of quasilinear stars in $mathrm{Ray}(V)$. They provide stratifications of $mathrm{Ray}(V)$ and, therefore, a finer convex analysis that helps better understand geometric properties.
{"title":"Stratifications of the ray space of a tropical quadratic form by Cauchy-Schwartz functions","authors":"Z. Izhakian, Manfred Knebusch","doi":"10.13001/ela.2022.6493","DOIUrl":"https://doi.org/10.13001/ela.2022.6493","url":null,"abstract":"Classes of an equivalence relation on a module $V$ over a supertropical semiring, called rays, carry the underlying structure of 'supertropical trigonometry' and thereby a version of convex geometry which is compatible with quasilinearity. In this theory, the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio, which gives rise to special characteristic functions, called CS-functions. These functions partite the ray space $mathrm{Ray}(V)$ into convex sets and establish the main tool for analyzing varieties of quasilinear stars in $mathrm{Ray}(V)$. They provide stratifications of $mathrm{Ray}(V)$ and, therefore, a finer convex analysis that helps better understand geometric properties.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46815883","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}
Given a graph $G$ with $mge 1$ edges, the non-backtracking spectral radius of $G$ is the spectral radius of its non-backtracking matrix $B(G)$ defined as the $2m times 2m$ matrix where each edge $uv$ is represented by two rows and two columns, one per orientation: $(u, v)$ and $(v, u)$, and the entry of $B(G)$ in row $(u, v)$ and column $(x,y)$ is given by $delta_{vx}(1-delta_{uy})$, with $delta_{ij}$ being the Kronecker delta. A tight upper bound is given for the non-backtracking spectral radius in terms of the spectral radius of the adjacency matrix and minimum degree, and those connected graphs that maximize the non-backtracking spectral radius are determined if the connectivity (edge connectivity, bipartiteness, respectively) is given.
{"title":"On the non-backtracking spectral radius of graphs","authors":"Hongying Lin, B. Zhou","doi":"10.13001/ela.2022.6507","DOIUrl":"https://doi.org/10.13001/ela.2022.6507","url":null,"abstract":"Given a graph $G$ with $mge 1$ edges, the non-backtracking spectral radius of $G$ is the spectral radius of its non-backtracking matrix $B(G)$ defined as the $2m times 2m$ matrix where each edge $uv$ is represented by two rows and two columns, one per orientation: $(u, v)$ and $(v, u)$, and the entry of $B(G)$ in row $(u, v)$ and column $(x,y)$ is given by $delta_{vx}(1-delta_{uy})$, with $delta_{ij}$ being the Kronecker delta. A tight upper bound is given for the non-backtracking spectral radius in terms of the spectral radius of the adjacency matrix and minimum degree, and those connected graphs that maximize the non-backtracking spectral radius are determined if the connectivity (edge connectivity, bipartiteness, respectively) is given.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43163865","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 linear dynamical systems (in continuous-time and discrete-time), we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.
{"title":"Hypocoercivity and hypocontractivity concepts for linear dynamical systems","authors":"F. Achleitner, A. Arnold, V. Mehrmann","doi":"10.13001/ela.2023.7531","DOIUrl":"https://doi.org/10.13001/ela.2023.7531","url":null,"abstract":"For linear dynamical systems (in continuous-time and discrete-time), we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49480087","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 a study of the eigenvalues of a complex square matrix with one variable nondiagonal entry expressed in polar form. Changing the angle of the variable entry while leaving the radius fixed generates an algebraic curve; as does the process of fixing an angle and varying the radius. The authors refer to these two curves as eigenvalue orbits and eigenvalue trajectories, respectively. Eigenvalue orbits and trajectories are orthogonal families of curves, and eigenvalue orbits are sets of eigenvalues from matrices with identical Gershgorin regions. Algebraic and geometric properties of both types of curves are examined. Features such as poles, singularities, and foci are discussed.
{"title":"On the eigenvalues of matrices with common Gershgorin regions","authors":"Anna Davis, P. Zachlin","doi":"10.13001/ela.2022.6025","DOIUrl":"https://doi.org/10.13001/ela.2022.6025","url":null,"abstract":"This paper is a study of the eigenvalues of a complex square matrix with one variable nondiagonal entry expressed in polar form. Changing the angle of the variable entry while leaving the radius fixed generates an algebraic curve; as does the process of fixing an angle and varying the radius. The authors refer to these two curves as eigenvalue orbits and eigenvalue trajectories, respectively. Eigenvalue orbits and trajectories are orthogonal families of curves, and eigenvalue orbits are sets of eigenvalues from matrices with identical Gershgorin regions. Algebraic and geometric properties of both types of curves are examined. Features such as poles, singularities, and foci are discussed.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47679961","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}
We obtain formulae for group inverses of matrices that are associated with a new class of digraphs obtained from stars. This new class contains both bipartite and non-bipartite graphs. Expressions for the group inverse of matrices corresponding to double star digraphs and the adjacency matrix of certain undirected multi-star graphs are also proven. A blockwise representation of the inverse or group inverse of the adjacency matrix of the Dutch windmill graph is presented.
{"title":"Group inverses of matrices associated with certain graph classes","authors":"J. McDonald, R. Nandi, K. Sivakumar","doi":"10.13001/ela.2022.6717","DOIUrl":"https://doi.org/10.13001/ela.2022.6717","url":null,"abstract":"We obtain formulae for group inverses of matrices that are associated with a new class of digraphs obtained from stars. This new class contains both bipartite and non-bipartite graphs. Expressions for the group inverse of matrices corresponding to double star digraphs and the adjacency matrix of certain undirected multi-star graphs are also proven. A blockwise representation of the inverse or group inverse of the adjacency matrix of the Dutch windmill graph is presented.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46221861","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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