Electronic Journal of Linear Algebra
Latest Publications


TOTAL DOCUMENTS

1146
(FIVE YEARS 171)

H-INDEX

20
(FIVE YEARS 2)

Published By University Of Wyoming Libraries

1081-3810

2022 ◽  
pp. 32-48
Author(s):  
Georgios Katsouleas ◽  
Vasiliki Panagakou ◽  
Panayiotis Psarrakos

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.


2022 ◽  
pp. 63-64
Author(s):  
Yaroslav Shitov

It is known that, for any real m-by-n matrix A of rank n-2, there is a rational m-by-n matrix which has rank n-2 and sign pattern equal to that of  A. We prove a more general result conjectured in the recent literature.  


2022 ◽  
pp. 49-62
Author(s):  
Jonathan Caalim ◽  
Yu-ichi Tanaka

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ 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.


Author(s):  
Sooyeong Kim

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.


Author(s):  
Ralph John De la Cruz ◽  
Eloise Misa

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 $C(S).$


2021 ◽  
Vol 37 ◽  
pp. 734-746
Author(s):  
Wai Leong Chooi ◽  
Yean Nee Tan

Let $n\geq 2$ and $1<k\leq n$ be integers. Let $S_n(\mathbb{F})$ be the linear space of $n\times n$ 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 $\psi(A)A=A\psi(A)$ for all rank $k$ matrices $A\in S_n(\mathbb{F})$ if and only if there exists a scalar $\lambda\in \mathbb{F}$ and an additive map $\mu:S_n(\mathbb{F})\rightarrow \mathbb{F}$ such that\[\psi(A)=\lambda A+\mu(A)I_n,\]for all $A\in S_n(\mathbb{F})$, 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.


2021 ◽  
Vol 37 ◽  
pp. 747-757
Author(s):  
Amy Yielding ◽  
Taylor Hunt ◽  
Joel Jacobs ◽  
Jazmine Juarez ◽  
Taylor Rhoton ◽  
...  

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.


2021 ◽  
Vol 37 ◽  
pp. 718-733
Author(s):  
Semra Ozturk

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.


2021 ◽  
Vol 37 ◽  
pp. 709-717
Author(s):  
Mustapha Aouchiche ◽  
Bilal A. Rather ◽  
Issmail El Hallaoui

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 Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.


2021 ◽  
Vol 37 ◽  
pp. 698-708
Author(s):  
Qinghong Zhang

The paper is devoted to the study of the maximal angle between the $5\times 5$ semidefinite matrix cone and $5\times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.


Sign in / Sign up

Export Citation Format

Share Document