Electronic Journal of Linear AlgebraCopyright (c) 2019 University of Wyoming All rights reserved.
https://repository.uwyo.edu/ela
Recent documents in Electronic Journal of Linear Algebraen-usSat, 18 May 2019 04:48:27 PDT3600The Determinant and Complex Gershgorin Circles
https://repository.uwyo.edu/ela/vol35/iss1/11
https://repository.uwyo.edu/ela/vol35/iss1/11Sat, 18 May 2019 04:45:28 PDT
Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.
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Florian Bünger et al.Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States
https://repository.uwyo.edu/ela/vol35/iss1/10
https://repository.uwyo.edu/ela/vol35/iss1/10Sat, 18 May 2019 04:45:23 PDT
A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.
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Nathaniel Johnston et al.Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials
https://repository.uwyo.edu/ela/vol35/iss1/9
https://repository.uwyo.edu/ela/vol35/iss1/9Mon, 22 Apr 2019 10:04:46 PDT
The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.
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Biswajit Das et al.Potentially Eventually Positive 2-generalized Star Sign Patterns
https://repository.uwyo.edu/ela/vol35/iss1/8
https://repository.uwyo.edu/ela/vol35/iss1/8Mon, 22 Apr 2019 10:04:33 PDT
A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually exponentially positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a nonnegative integer $t_{0}$ such that $e^{tA}=\sum_{k=0}^{\infty}\frac{t^{k}A^{k}}{k!}>0$ for all $t\geq t_{0}$. Identifying necessary and sufficient conditions for an $n$-by-$n$ sign pattern to be potentially eventually positive (respectively, potentially eventually exponentially positive), and classifying these sign patterns are open problems. In this article, the potential eventual positivity of the $2$-generalized star sign patterns is investigated. All the minimal potentially eventually positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually positive $2$-generalized star sign patterns are classified. As an application, all the minimal potentially eventually exponentially positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually exponentially positive $2$-generalized star sign patterns are classified.
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Yu Ber-Lin et al.Jordan Triple Product Homomorphisms on Triangular Matrices to and from Dimension One
https://repository.uwyo.edu/ela/vol33/iss1/14
https://repository.uwyo.edu/ela/vol33/iss1/14Thu, 11 Apr 2019 11:53:29 PDT
A map $\Phi$ is a Jordan triple product (JTP for short) homomorphism whenever $\Phi(A B A)= \Phi(A) \Phi(B) \Phi(A)$ for all $A,B$. We study JTP homomorphisms on the set of upper triangular matrices $\mathcal{T}_n(\mathbb{F})$, where $\Ff$ is the field of real or complex numbers. We characterize JTP homomorphisms $\Phi: \mathcal{T}_n(\mathbb{C}) \to \mathbb{C}$ and JTP homomorphisms $\Phi: \mathbb{F} \to \mathcal{T}_n(\mathbb{F})$. In the latter case we consider continuous maps and the implications of omitting the assumption of continuity.
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Damjana Kokol Bukovsek et al.Unions of a clique and a co-clique as star complements for non-main graph eigenvalues
https://repository.uwyo.edu/ela/vol35/iss1/7
https://repository.uwyo.edu/ela/vol35/iss1/7Thu, 28 Mar 2019 19:28:19 PDT
Graphs consisting of a clique and a co-clique, both of arbitrary size, are considered in the role of star complements for an arbitrary non-main eigenvalue. Among other results, the sign of such a eigenvalue is discussed, the neigbourhoods of star set vertices are described, and the parameters of all strongly regular extensions are determined. It is also proved that, unless in a specified special case, if the size of a co-clique is fixed then there is a finite number of possibilities for our star complement and the corresponding non-main eigenvalue. Numerical data on these possibilities is presented.
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Zoran StanicAE Regularity of Interval Matrices
https://repository.uwyo.edu/ela/vol33/iss1/13
https://repository.uwyo.edu/ela/vol33/iss1/13Thu, 28 Mar 2019 19:24:10 PDT
Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ∀∃- quantification. The paper deals with the problem of what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, a concept of AE regularity is introduced, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. A characterization of AE regularity is discussed, and also various classes of matrices that are implicitly AE regular are investigated. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. Eventually, there are also stated open problems related to computational complexity and characterization of AE regularity.
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Milan HladíkCondensed Forms for Linear Port-Hamiltonian Descriptor Systems
https://repository.uwyo.edu/ela/vol35/iss1/6
https://repository.uwyo.edu/ela/vol35/iss1/6Sat, 23 Mar 2019 10:57:10 PDT
Motivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by $\mu\leq1$.
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Lena ScholzBrauer's theorem and nonnegative matrices with prescribed diagonal entries
https://repository.uwyo.edu/ela/vol35/iss1/5
https://repository.uwyo.edu/ela/vol35/iss1/5Fri, 15 Feb 2019 13:17:52 PST
The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.
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Ricardo L. Soto et al.Diagonal Sums of Doubly Substochastic Matrices
https://repository.uwyo.edu/ela/vol35/iss1/4
https://repository.uwyo.edu/ela/vol35/iss1/4Fri, 15 Feb 2019 13:17:33 PST
Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.
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Lei Cao et al.In-sphere property and reverse inequalities for matrix means
https://repository.uwyo.edu/ela/vol35/iss1/3
https://repository.uwyo.edu/ela/vol35/iss1/3Sat, 09 Feb 2019 17:41:40 PST
The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.
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Trung Hoa Dinh et al.Surjective Additive Rank-1 Preservers on Hessenberg Matrices
https://repository.uwyo.edu/ela/vol35/iss1/2
https://repository.uwyo.edu/ela/vol35/iss1/2Sat, 09 Feb 2019 17:41:29 PST
Let $H_{n}(\mathbb{F})$ be the space of all $n\times n$ upper Hessenberg matrices over a field~$\mathbb{F}$, where $n$ is a positive integer greater than two. In this paper, surjective additive maps preserving rank-$1$ on $H_{n}(\mathbb{F})$ are characterized.
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PRATHOMJIT KHACHORNCHAROENKUL et al.Solving the Sylvester Equation AX-XB=C when $\sigma(A)\cap\sigma(B)\neq\emptyset$
https://repository.uwyo.edu/ela/vol35/iss1/1
https://repository.uwyo.edu/ela/vol35/iss1/1Tue, 05 Feb 2019 21:45:04 PST
The method for solving the Sylvester equation $AX-XB=C$ in complex matrix case, when $\sigma(A)\cap\sigma(B)\neq \emptyset$, by using Jordan normal form is given. Also, the approach via Schur decomposition is presented.
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Nebojša Č. DinčićResolution Of Conjectures Related To Lights Out! And Cartesian Products
https://repository.uwyo.edu/ela/vol34/iss1/51
https://repository.uwyo.edu/ela/vol34/iss1/51Wed, 16 Jan 2019 21:12:33 PST
Lights Out!\ is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out!\ on Cartesian products of graphs are resolved.
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Bryan A. Curtis et al.On the Interval Generalized Coupled Matrix Equations
https://repository.uwyo.edu/ela/vol34/iss1/50
https://repository.uwyo.edu/ela/vol34/iss1/50Wed, 16 Jan 2019 21:12:23 PST
In this work, the interval generalized coupled matrix equations \begin{equation*} \sum_{j=1}^{p}{{\bf{A}}_{ij}X_{j}}+\sum_{k=1}^{q}{Y_{k}{\bf{B}}_{ik}}={\bf{C}}_{i}, \qquad i=1,\ldots,p+q, \end{equation*} are studied in which ${\bf{A}}_{ij}$, ${\bf{B}}_{ik}$ and ${\bf{C}}_{i}$ are known real interval matrices, while $X_{j}$ and $Y_{k}$ are the unknown matrices for $j=1,\ldots,p$, $k=1,\ldots,q$ and $i=1,\ldots,p+q$. This paper discusses the so-called AE-solution sets for this system. In these types of solution sets, the elements of the involved interval matrices are quantified and all occurrences of the universal quantifier $\forall$ (if any) precede the occurrences of the existential quantifier $\exists$. The AE-solution sets are characterized and some sufficient conditions under which these types of solution sets are bounded are given. Also some approaches are proposed which include a numerical technique and an algebraic approach for enclosing some types of the AE-solution sets.
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Marzieh Dehghani-MadisehA note on linear preservers of semipositive and minimally semipositive matrices
https://repository.uwyo.edu/ela/vol34/iss1/49
https://repository.uwyo.edu/ela/vol34/iss1/49Tue, 08 Jan 2019 11:59:04 PST
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this short note, the structure of linear maps which preserve the set of all semipositive/minimally semipositive matrices is studied. An open problem is solved, and some ambiguities in the article [J. Dorsey, T. Gannon, N. Jacobson, C.R. Johnson and M. Turnansky. Linear preservers of semi-positive matrices. {\em Linear and Multilinear Algebra}, 64:1853--1862, 2016.] are clarified.
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Projesh Nath Choudhury et al.Vector Cross Product Differential and Difference Equations in R^3 and in R^7
https://repository.uwyo.edu/ela/vol34/iss1/48
https://repository.uwyo.edu/ela/vol34/iss1/48Tue, 08 Jan 2019 11:58:51 PST
Through a matrix approach of the $2$-fold vector cross product in $\mathbb{R}^3$ and in $\mathbb{R}^7$, some vector cross product differential and difference equations are studied. Either the classical theory or convenient Drazin inverses, of elements belonging to the class of index $1$ matrices, are applied.
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Patrícia D. Beites et al.Gershgorin type sets for eigenvalues of matrix polynomials
https://repository.uwyo.edu/ela/vol34/iss1/47
https://repository.uwyo.edu/ela/vol34/iss1/47Wed, 26 Dec 2018 15:32:09 PST
New localization results for polynomial eigenvalue problems are obtained, by extending the notions of the Gershgorin set, the generalized Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set to the case of matrix polynomials.
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Christina Michailidou et al.Determinantal Properties of Generalized Circulant Hadamard Matrices
https://repository.uwyo.edu/ela/vol34/iss1/46
https://repository.uwyo.edu/ela/vol34/iss1/46Wed, 26 Dec 2018 15:31:57 PST
The derivation of analytical formulas for the determinant and the minors of a given matrix is in general a difficult and challenging problem. The present work is focused on calculating minors of generalized circulant Hadamard matrices. The determinantal properties are studied explicitly, and generic theorems specifying the values of all the minors for this class of matrices are derived. An application of the derived formulae to an interesting problem of numerical analysis, the growth problem, is also presented.
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Marilena Mitrouli et al.On the Condition Number Theory of the Equality Constrained Indefinite Least Squares Problem
https://repository.uwyo.edu/ela/vol34/iss1/45
https://repository.uwyo.edu/ela/vol34/iss1/45Wed, 26 Dec 2018 15:31:43 PST
In this paper, within a unified framework of the condition number theory, the explicit expression of the \emph{projected} condition number of the equality constrained indefinite least squares problem is presented. By setting specific norms and parameters, some widely used condition numbers, like the normwise, mixed and componentwise condition numbers follow as its special cases. Considering practical applications and computation, some new compact forms or upper bounds of the projected condition numbers are given to improve the computational efficiency. The new compact forms are of particular interest in calculating the exact value of the 2-norm projected condition numbers. When the equality constrained indefinite least squares problem degenerates into some specific least squares problems, our results give some new findings on the condition number theory of these specific least squares problems. Numerical experiments are given to illustrate our theoretical results.
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Shaoxin Wang et al.