Matrix Classes Research Articles

Characterizations of complex P-matrices

A P-matrix is a square matrix all of whose principal minors are positive. The characterization of real P-matrices as matrices that do not reverse the sign of any nonzero real vector is generalized to complex P-matrices by associating them with the reflection of complex vectors. This prompts the extension of other P-matrix properties and related real matrix classes to the complex field. In particular, semipositivity of real P-matrices is generalized to complex P-matrices. Principal pivot transforms and Cayley transforms of complex P-matrices are also considered.

Schröder–Catalan Matrix and Compactness of Matrix Operators on Its Associated Sequence Spaces

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented.

On the $q$-Cesaro bounded double sequence space

In this article, the new sequence space $\tilde{\mathcal{M}}_u^q$ is acquainted, described as the domain of the 4d (4-dimensional) $q$-Cesaro matrix operator, which is the $q$-analogue of the first order 4d Cesaro matrix operator, on the space of bounded double sequences. In the continuation of the study, the completeness of the new space is given, and the inclusion relation related to the space is presented. In the last two parts, the duals of the space are determined, and some matrix classes are acquired.

New Integrated and Differentiated Sequence Spaces ∫ b_p^(r,s) and db_p^(r,s)

In this work, we construct new sequence spaces by combining the integrated and differentiated sequence spaces with the binomial matrix. Firstly, we provide information about basic matters such as sequence spaces and matrix domain. Subsequently we briefly summarize some sequence spaces generated by the binomial matrix. Thereafter, we define the integrated and differentiated sequence spaces and establish the new sequence spaces. Afterwards, we examine some properties and the inclusion relations of these new sequence spaces. We also determine the α, β and γ-duals of the integrated and differentiated sequence spaces. Finally, we characterize some matrix classes associated with the new sequence spaces.

Characterizations of some new classes of four-dimensional matrices and dual spaces on the double spaces of Cesàro means

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces |C_{1,1}|_{k} defined by combining the first order Cesàro means with the concept of absolute summability k≥1. Beside this, we determine the α-dual of the space |C_{1,1}|₁ and the β(bp)- and γ-duals of the spaces |C_{1,1}|_{k} for k≥1. Finally, we characterize some new four-dimensional matrix classes (|C_{1,1}|_{k},υ), (|C_{1,1}|₁,υ), (|C_{1,1}|₁,L_{k}), (|C_{1,1}|_{k},L_{u}), (L_{u},|C_{1,1}|_{k}) and (L_{k},|C_{1,1}|₁), where υ∈{M_{u},C_{bp}} for 1≤k

Spectral arbitrariness for trees fails spectacularly

Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.

Networks of reinforced stochastic processes: A complete description of the first-order asymptotics

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions for the emergence of some form of almost sure asymptotic synchronization. Specifically, we identify three regimes: the first involves complete synchronization, where all processes converge towards the same random variable; the second exhibits almost sure convergence of the system, but no form of synchronization subsists; and the third reveals a scenario where there is almost sure asymptotic synchronization within the cyclic classes of the interaction matrix, together with an asymptotic periodic behavior among these classes.

Erratum: Properties of the Solution Set of Absolute Value Equations and the Related Matrix Classes

Erratum: Properties of the Solution Set of Absolute Value Equations and the Related Matrix Classes

Matrix Transformation and Compactness on q-Catalan Sequence Spaces

This article intends to develop q-Catalan sequence spaces l p ( C ( q ) ) and l ∞ ( C ( q ) ) due to q-Catalan matrix C ( q ) in l p and l ∞ , respectively. Apart from obtaining some basic topological properties and Schauder basis, we compute α-, β-, and γ-duals of the spaces l p ( C ( q ) ) and l ∞ ( C ( q ) ) . We state and prove a theorem that characterize certain matrix classes (X, Y), where X is either of the space l p ( C ( q ) ) or l ∞ ( C ( q ) ) and Y ∈ { l ∞ , c 0 , c , l 1 } . The final section is devoted to determination of certain conditions by which a matrix operator becomes compact.

Mersenne Matrix Operator and Its Application in $p-$Summable Sequence Space

In this study, it is introduced the regular Mersenne matrix operator which is obtained by using Mersenne numbers and examined sequence spaces described as the domain of this matrix in the space of $p$-summable sequences for $1\leq p \leq \infty$. After that, it investigated some properties and inclusion relations, established the Schauder basis, and stated $\alpha-$, $\beta-$, and $\gamma-$duals of the aforementioned spaces. Additionally, it is characterized by the matrix classes from newly described spaces to classical sequence spaces. Finally, we studied the compactness of matrix operators on related sequence spaces.

Matrices in [formula omitted] with quadratic minimal polynomial

By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of n×n matrices over Fq[T] with minimum polynomial p(X), where p is an nth degree polynomial, irreducible over Fq[T]. In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for p(X)=X2−ΓX−Δ∈Fq[T][X].

Compact operators on the new Motzkin sequence spaces

<p>This study aims to construct the BK-spaces $ \ell_p(\mathcal{M}) $ and $ \ell_{\infty}(\mathcal{M}) $ as the domains of the conservative Motzkin matrix $ \mathcal{M} $ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $ \alpha $-, $ \beta $-, and $ \gamma $-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.</p>

On The Computational Properties of 3-Cyclic and 4-Cyclic Refined Matrices and the Diagonalization Algorithm

This paper is concerned with studying the matrix computations of 3-cyclic refined neutrosophic matrices and 4-cyclic refined neutrosophic matrices with 3cyclic4-cyclic real entries, where we introduce a novel method to compute eigenvalues and vectors of these matrix classes. Also, we provide a novel algorithm for diagonalization these matrices and to determine whether an n-cyclic refined matrix is diagonalizable or not for n=3, 4.

MRI radiomics based machine learning model of the periaqueductal gray matter in migraine patients.

<p>The aim of the study was to investigate the question: Can MRI radiomics analysis of the periaqueductal gray region elucidate the pathophysiological mechanisms underlying various migraine subtypes, and can a machine learning model using these radiomics features accurately differentiate between migraine patients and healthy individuals, as well as between migraine subtypes, including atypical cases with overlapping symptoms?</p>. <p>The study analyzed initial MRI images of individuals taken after their first migraine diagnosis, and additional MRI scans were acquired from healthy subjects. Radiomics modeling was applied to analyze all the MRI images in the periaqueductal gray region. The dataset was randomized, and oversampling was used if there was class imbalance between groups. The optimal algorithm-based feature selection method was employed to select the most important 5-10 features to differentiate between the two groups. The classification performance of AI algorithms was evaluated using receiver operating characteristic analysis to calculate the area under the curve, classification accuracy, sensitivity, and specificity values. Participants were required to have a confirmed diagnosis of either episodic migraine, probable migraine, or chronic migraine. Patients with aura, those who used migraine-preventive medication within the past six months, or had chronic illnesses, psychiatric disorders, cerebrovascular conditions, neoplastic diseases, or other headache types were excluded from the study. Additionally, 102 healthy subjects who met the inclusion and exclusion criteria were included.&nbsp;</p>. <p>The algorithm-based information gain method for feature reduction had the best performance among all methods, with the first-order, gray-level size zone matrix, and gray-level co-occurrence matrix classes being the dominant feature classes. The machine learning model correctly classified 82.4% of migraine patients from healthy subjects. Within the migraine group, 74.1% of the episodic migraine-probable migraine patients and 90.5% of the chronic migraine patients were accurately classified. No significant difference was found between probable migraine and episodic migraine patients in terms of the periaqueductal gray region radiomics features. The kNN algorithm showed the best performance for classifying episodic migraine-probable migraine subtypes, while the Random Forest algorithm demonstrated the best performance for classifying the migraine group and chronic migraine subtype.</p>. <p>A radiomics-based machine learning model, utilizing standard MR images obtained during the diagnosis and follow-up of migraine patients, shows promise not only in aiding migraine diagnosis and classification for clinical approach, but also in understanding the neurological mechanisms underlying migraines.&nbsp;</p>.

Four New Sequence Spaces Obtained from the Domain of Quadruple Band Matrix Operator

In this work, we construct the sequence spaces $c_{0}(Q)$, $c(Q)$, $\ell_{\infty}(Q)$ and $\ell_{p}(Q)$ derived by the domain of quadruple band matrix, which generalizes the matrices $\Delta^{3}$, $B(r,s,t)$, $\Delta^{2}$, $B(r,s)$, $\Delta$, where $\Delta^{3}$, $B(r,s,t)$, $\Delta^{2}$, $B(r,s)$ and $\Delta$ are called third order difference, triple band, second order difference, double band and difference matrix, in turn. Also, we investigate some topological properties and some inclusion relations related to those spaces. Furthermore, we give the Schauder basis of the spaces $c_{0}(Q)$, $c(Q)$ and $\ell_{p}(Q)$, and determine $\alpha-\beta-$ and $\gamma-$duals of those spaces. Lastly, we characterize some matrix classes related to some of those spaces.

A New Paranormed Sequence Space Defined by Regular Bell Matrix

This paper aims to construct a new paranormed sequence space by the aid of a regular matrix of Bell numbers. As well, its special duals such as α−,β−,γ− duals are presented and Schauder basis is determined. Moreover, certain matrix classes for this space are characterized.

Two New Matrix Classes Related to the CMP Inverse: CMP and Co-CMP Matrices

This paper focuses on two new matrix classes related to the CMP inverse of a square matrix, called the CMP and co-CMP matrices. Some of their characterizations are identified based on the core-EP decomposition and subspace operations. And, we show an equivalence relationship between the co-CMP matrix set and the intersection of the core matrix set and the co-EP matrix set. The nonsingularity of other matrices closely associated with the co-CMP matrices is considered, and their inverses are presented in a decomposition form.

Sufficient Matrices: Properties, Generating and Testing

This paper investigates various aspects of sufficient matrices, one of the most relevant matrix classes introduced in connection with linear complementarity problems. We summarize the most important theoretical results and properties related to sufficient matrices. Based on these, we propose different construction rules that can be used to generate new matrices that belong to this class. A nonnegative number can be assigned to each sufficient matrix, which is called its handicap and works as a measure of sufficiency. The handicap plays a crucial role in proving convergence and complexity results for interior point algorithms for linear complementarity problems. For a particular sufficient matrix, called Csizmadia’s matrix, we give the exact value of the handicap, which is exponential in the size of the matrix. Another important topic that we address is deciding whether a matrix is sufficient. Tseng proved in 2000 that this decision problem is co-NP hard. We investigate three different algorithms for determining the sufficiency of a given matrix: Väliaho’s algorithm, a linear programming-based algorithm, and an algorithm that facilitates nonlinear programming reformulations of the definition of sufficiency. We tested the efficiency of these methods on our recently launched benchmark data set that consists of four different sets of matrices. In this paper, we give the description and most important properties of the benchmark set, which can be used in the future to compare the performance of different interior point algorithms for linear complementarity problems.

A two‐step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

AbstractFor solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems.

On Some Special Matrix Classes Useful in Optimization and Game Theory

The well-known problem of checking copositivity is equivalent to solving a quadratic optimization problem over the standard simplex. This is an object of research for a long time. This paper presents some results related to the well-known matrix classes like copositive, semimonotone matrices discussed in the literature for many years. We relate some important theorems of alternatives with the class of copositive-plus and copositive-star matrices. Finally, we revisit the classes of almost copositive, semimonotone matrices and obtain some new results which are useful in optimization theory. Finally, we present some game theoretic characterizations of these special class of matrices.