scholarly journals Left and Right Inverse Eigenpairs Problem forκ-Hermitian Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Fan-Liang Li ◽  
Xi-Yan Hu ◽  
Lei Zhang
Keyword(s):  
Approximate Solution ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Optimal Approximate Solution ◽  
Hermitian Matrices ◽  
Equivalent Problem ◽  
Approximate Problem ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Product Of Matrices

Left and right inverse eigenpairs problem forκ-hermitian matrices and its optimal approximate problem are considered. Based on the special properties ofκ-hermitian matrices, the equivalent problem is obtained. Combining a new inner product of matrices, the necessary and sufficient conditions for the solvability of the problem and its general solutions are derived. Furthermore, the optimal approximate solution and a calculation procedure to obtain the optimal approximate solution are provided.

An inverse eigenproblem for generalized reflexive matrices with normal $k+1$-potencies

2016 ◽  
Vol 31 ◽  
pp. 100-119 ◽  
Author(s):  
Wei-Ru Xu ◽  
Guo-Liang Chen
Keyword(s):  
Approximate Solution ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Frobenius Norm ◽  
Optimal Approximate Solution ◽  
General Representation ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Left And Right Eigenvalues ◽  
Norm Approximation

Let $P,~Q\in\mathbb{C}^{n\times n}$ be two normal $\{k+1\}$-potent matrices, i.e., $PP^{*}=P^{*}P,~P^{k+1}=P$, $QQ^{*}=Q^{*}Q,~Q^{k+1}=Q$, $k\in\mathbb{N}$. A matrix $A\in\mathbb{C}^{n\times n}$ is referred to as generalized reflexive with two normal $\{k+1\}$-potent matrices $P$ and $Q$ if and only if $A=PAQ$. The set of all $n\times n$ generalized reflexive matrices which rely on the matrices $P$ and $Q$ is denoted by $\mathcal{GR}^{n\times n}(P,Q)$. The left and right inverse eigenproblem of such matrices ask from us to find a matrix $A\in\mathcal{GR}^{n\times n}(P,Q)$ containing a given part of left and right eigenvalues and corresponding left and right eigenvectors. In this paper, first necessary and sufficient conditions such that the problem is solvable are obtained. A general representation of the solution is presented. Then an expression of the solution for the optimal Frobenius norm approximation problem is exploited. A stability analysis of the optimal approximate solution, which has scarcely been considered in existing literature, is also developed.

scholarly journals Structure-preserving Diagonalization of Matrices in Indefinite Inner Product Spaces

2020 ◽  
Vol 36 (36) ◽  
pp. 21-37
Author(s):  
Philip Saltenberger
Keyword(s):  
Sufficient Conditions ◽  
Inner Product ◽  
Product Spaces ◽  
Hermitian Matrices ◽  
Additive Decomposition ◽  
Indefinite Inner Product ◽  
Structure Preserving ◽  
Inner Product Spaces ◽  
Structured Matrix ◽  
Necessary And Sufficient

In this work some results on the structure-preserving diagonalization of selfadjoint and skewadjoint matrices in indefinite inner product spaces are presented. In particular, necessary and sufficient conditions on the symplectic diagonalizability of (skew)-Hamiltonian matrices and the perplectic diagonalizability of per(skew)-Hermitian matrices are provided. Assuming the structured matrix at hand is additionally normal, it is shown that any symplectic or perplectic diagonalization can always be constructed to be unitary. As a consequence of this fact, the existence of a unitary, structure-preserving diagonalization is equivalent to the existence of a specially structured additive decomposition of such matrices. The implications of this decomposition are illustrated by several examples.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

scholarly journals Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Yalu Li ◽  
Wenhui Dou ◽  
Haitao Li ◽  
Xin Liu
Keyword(s):  
Matrix Method ◽  
Finite Automata ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Form ◽  
Controllability Matrix ◽  
Necessary And Sufficient ◽  
Semitensor Product Of Matrices ◽  
Product Of Matrices ◽  
Inputs And Outputs

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Finally, an illustrative example is given to support the obtained new results.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1981 ◽  
Vol 33 (1) ◽  
pp. 229-246 ◽  
Author(s):  
Calvin D. Ahlbrandt ◽  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Dimensional Vector ◽  
Selfadjoint Extension ◽  
Image Position ◽  
Vector Valued Function ◽  
Necessary And Sufficient ◽  
Vector Valued

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

2016 ◽  
Vol 6 (1) ◽  
pp. 42-59 ◽  
Author(s):  
Wei-Ru Xu ◽  
Guo-Liang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Structural Model ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Frobenius Norm ◽  
Structured Matrix ◽  
Inverse Eigenvalue ◽  
Left And Right ◽  
Necessary And Sufficient

AbstractGeneralised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation where M, D, G, K are given matrices with appropriate sizes and x is a column vector, we design a new vibrating structure mode. This mode can be discretised as the left and right inverse eigenvalue problem of a certain structured matrix. When the structured matrix is generalised centrohermitian, we discuss its left and right inverse eigenvalue problem with a submatrix constraint, and then get necessary and sufficient conditions such that the problem is solvable. A general representation of the solutions is presented, and an analytical expression for the solution of the optimal approximation problem in the Frobenius norm is obtained. Finally, the corresponding algorithm to compute the unique optimal approximate solution is presented, and we provide an illustrative numerical example.

scholarly journals Best Proximity Points for Some Classes of Proximal Contractions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Maryam A. Alghamdi ◽  
Naseer Shahzad ◽  
Francesca Vetro
Keyword(s):  
Approximate Solution ◽  
Nonlinear Programming ◽  
Programming Problem ◽  
Iterative Algorithm ◽  
Best Proximity Point ◽  
Sufficient Conditions ◽  
Nonlinear Programming Problem ◽  
Optimal Approximate Solution ◽  
Unique Point ◽  
Banach’S Contraction Principle

Given a self-mapping and a non-self-mapping , the aim of this work is to provide sufficient conditions for the existence of a unique point , calledg-best proximity point, which satisfies . In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function , thereby getting an optimal approximate solution to the equation . An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach's contraction principle to the case of non-self-mappings.

scholarly journals Characterizations of variational convergence of bifunctions defined on products of two sets

2020 ◽  
Vol 3 (SI3) ◽  
pp. First
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Basic Property ◽  
Specific Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Variational Convergence ◽  
Product Spaces ◽  
The Third ◽  
Left And Right ◽  
Definition Of ◽  
Necessary And Sufficient

We present definitions of types of variational convergence of finite-valued bifunctions defined on rectangular domains and establish characterizations of these convergences. In the introduction, we present the origins of the research on variational convergence and then we lead to the specific problem of this paper. The content of the paper consists of 3 parts: variational convergance of fucntion; variational convergance of bifunction; and characterizations of variational convergence of bifunction, this part is the main results of this paper. In section 2, we presented the definition of epi convergence and presented a basic property problem that will be used to extend and develop the next two sections. In section 3, we start to present a new definition, the definition of convergence epi / hypo, minsup and maxinf. To clearly understand of these new definitions we have provided comments (remarks) and some examples which reader can check these definitions. The above contents serve the main result of this paper will apply in part 4. Now, we will explain more detail for this part as follows. Firstly, variational convergence of bifunctions is characterized by the epi- and hypo-convergence of related unifunctions, which are slices sup- and inf-projections. The second characterization expresses the equivalence of variational convergence of bifunctions and the same convergence of the so-called proper bifunctions defined on the whole product spaces. In the third one, the geometric reformulation, we establish explicitly the interval of all the limits by computing formulae of the left- and right-end limit bifunctions, and this is necessary and sufficient conditions of the sequence bifunctions to attain epi / hypo, minsup and maxinf convergence.

scholarly journals A Note on the Observability of Temporal Boolean Control Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Wenping Shi ◽  
Bo Wu ◽  
Jing Han
Keyword(s):  
Boolean Network ◽  
Sufficient Conditions ◽  
Boolean Networks ◽  
Control Networks ◽  
Linear Dynamic Systems ◽  
Regulatory Interactions ◽  
Boolean Control Network ◽  
Boolean Network Model ◽  
Necessary And Sufficient ◽  
Product Of Matrices

Temporal Boolean network is a generalization of the Boolean network model that takes into account the time series nature of the data and tries to incorporate into the model the possible existence of delayed regulatory interactions among genes. This paper investigates the observability problem of temporal Boolean control networks. Using the semi tensor product of matrices, the temporal Boolean networks can be converted into discrete time linear dynamic systems with time delays. Then, necessary and sufficient conditions on the observability via two kinds of inputs are obtained. An example is given to illustrate the effectiveness of the obtained results.

scholarly journals Left and Right Magnifying Elements in Generalized Semigroups of Transformations by Using Partitions of a Set

2018 ◽  
Vol 11 (3) ◽  
pp. 580-588
Author(s):  
Ronnason Chinram ◽  
Pattarawan Petchkaew ◽  
Samruam Baupradist
Keyword(s):  
Linear Transformation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Proper Subset ◽  
Transformation Semigroups ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Composition Of Functions

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let X be a nonempty set and T(X) be the semigroup of all transformation from X into itself under the composition of functions. For a partition P = {X_α | α ∈ I} of the set X, let T(X,P) = {f ∈ T(X) | (X_α)f ⊆ X_α for all α ∈ I}. Then T(X,P) is a subsemigroup of T(X) and if P = {X}, T(X,P) = T(X). Our aim in this paper is to give necessary and sufficient conditions for elements in T(X,P) to be left or right magnifying. Moreover, we apply those conditions to give necessary and sufficient conditions for elements in some generalized linear transformation semigroups.

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