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For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3 × 3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

This paper focuses on algorithms for the truncated Takagi factorization of complex symmetric matrices. The problem is formulated as a Riemannian optimization problem on a complex Stiefel manifold and then is converted into a real Riemannian optimization problem on the intersection of the real Stiefel manifold and the quasi-symplectic set. The steepest descent, the Riemannian nonmonotone conjugate gradient, Newton, and hybrid methods are used for solving the problem and they are compared in their performance for the optimization task. Numerical experiments are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem is to determine when a multiset of n real numbers is the spectrum of a symmetric nonnegative $ n \times n $ n × n matrix. The problem is solved only for $ n\leq 4 $ n ≤ 4. In this article, the authors examine the set of possible spectra for nonnegative symmetric $ 6\times 6 $ 6 × 6 matrices, dividing the set into points that are realizable, points that are unrealizable by known necessary conditions, and points that satisfy all known necessary conditions, but their realizability remains unknown. [ABSTRACT FROM AUTHOR]

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the λ -pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (1) and T 3 (2) for the functions in this family. Further, we investigate several special cases and consequences of our results. [ABSTRACT FROM AUTHOR]

Let a = (a i) i ≥ 1 be a sequence in a field F , and f : F × F → F be a function such that f (a i , a i) ≠ 0 for all i ≥ 1. For any tournament T over [ n ] , consider the n × n symmetric matrix M T with zero diagonal whose (i , j) th entry (for i < j) is f (a i , a j) if i → j in T , and f (a j , a i) if j → i in T. It is known [3] that if T is a uniformly random tournament over [ n ] , then rank (M T) ≥ (1 2 − o (1)) n with high probability when char (F) ≠ 2 and f is a linear function. In this paper, we investigate the other extremal question: how low can the ranks of such matrices be? We work with sequences a that take only two distinct values, so the rank of any such n × n matrix is at least n / 2. First, we show that the rank of any such matrix depends on whether an associated bipartite graph has certain eigenvalues of high multiplicity. Using this, we show that if f is linear, then there are real matrices M T (f ; a) of rank at most n 2 + O (1). For rational matrices, we show that for each ε > 0 we can find a sequence a (ε) for which there are matrices M T (f ; a) of rank at most (1 2 + ε) n + O (1). These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than ⌊ 3 n / 2 ⌋ − 2 for n ≤ 15 , which improves the previously best known bound given in [5]. [ABSTRACT FROM AUTHOR]

Motivated by applications in matrix constructions used in the inverse eigenvalue problem for graphs, we study a concept of genericity for eigenvectors associated with a given spectrum and a connected graph. This concept generalizes the established notion of a nowhere-zero eigenbasis. Given any simple graph G on n vertices and any spectrum with no multiple eigenvalues, we show that the family of eigenbases for symmetric matrices with this graph and spectrum is generic, strengthening a result of Monfared and Shader. We illustrate applications of this result by constructing new achievable ordered multiplicity lists for partial joins of graphs and providing several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues. [ABSTRACT FROM AUTHOR]

This article focuses on model-based clustering of subjects based on the shared relationships of subject-specific networks and covariates in scenarios when there are differences in the relationship between networks and covariates for different groups of subjects. It is also of interest to identify the network nodes significantly associated with each covariate in each cluster of subjects. To address these methodological questions, we propose a novel nonparametric Bayesian mixture modeling framework with an undirected network response and scalar predictors. The symmetric matrix coefficients corresponding to the scalar predictors of interest in each mixture component involve low-rankness and group sparsity within the low-rank structure. While the low-rank structure in the network coefficients adds parsimony and computational efficiency, the group sparsity within the low-rank structure enables drawing inference on network nodes and cells significantly associated with each scalar predictor. Being a principled Bayesian mixture modeling framework, our approach allows model-based identification of the number of clusters, offers clustering uncertainty in terms of the co-clustering matrix and presents precise characterization of uncertainty in identifying network nodes significantly related to a predictor in each cluster. Empirical results in various simulation scenarios illustrate substantial inferential gains of the proposed framework in comparison with competitors. Analysis of a real brain connectome dataset using the proposed method provides interesting insights into the brain regions of interest (ROIs) significantly related to creative achievement in each cluster of subjects. shows the convergence rate for the posterior predictive density of the proposed model, additional simulation examples with model misspecification, full conditional distributions to run the Markov chain Monte Carlo (MCMC) algorithm and also presents traceplots for various model parameters to demonstrate convergence of the MCMC algorithm. [ABSTRACT FROM AUTHOR]

We propose a robust randomized indicator method for the reliable detection of eigenvalue existence within an interval for symmetric matrices A. An indicator tells the eigenvalue existence based on some statistical norm estimators for a spectral projector. Previous work on eigenvalue indicators relies on a threshold which is empirically chosen, thus often resulting in under or over detection. In this paper, we use rigorous statistical analysis to guide the design of a robust indicator. Multiple randomized estimators for a contour integral operator in terms of A are analyzed. In particular, when A has eigenvalues inside a given interval, we show that the failure probability (for the estimators to return very small estimates) is extremely low. This enables to design a robust rejection indicator based on the control of the failure probability. We also give a prototype framework to illustrate how the indicator method may be applied numerically for eigenvalue detection and may potentially serve as a new way to design randomized symmetric eigenvalue solvers. Unlike previous indicator methods that only detect eigenvalue existence, the framework also provides a way to find eigenvectors with little extra cost by reusing computations from indicator evaluations. Extensive numerical tests show the reliability of the eigenvalue detection in multiple aspects. [ABSTRACT FROM AUTHOR]

Solar photovoltaic (PV) system generates low power because of nonuniform sun irradiation compared to standard test conditions. Partial shading conditions (PSCs) can arise from various factors, with passing clouds, trees, telecommunication towers, bird droppings, and similar elements being among the most prevalent causes. Under the shading scenario, multiple global and local maxima power points exist on the P–V curves. Herein, the chess game rules for the "Rook" icon are followed to attain the optimal placement of integer numbers (1–9) in the 9 × 9 size game puzzle matrix. The integer number‐based game puzzles such as Su‐Do‐Ku, symmetric matrix, and proposed Rook's movement‐game theory (RMGT) to rearrange the PV modules in arrays are investigated and compared with the standard arrangements such as series‐parallel, total‐cross‐tied during PSCs. The suggested RMGT puzzle‐based arrangement is able to achieve higher power at global maximum power point, performance enhancement, fill factor, and low power loss. As a result, the experimental study proves the superiority of RMGT configuration compared to conventional methodologies under the shading scenarios. The proposed RMGT‐based PV array is tested and obtained the performance higher side as, 63.27ImVm$\left(63.27 I\right)_{m}V_{m}$, 64.17ImVm$\left(64.17 I\right)_{m}V_{m}$, 58.95ImVm$\left(58.95 I\right)_{m}V_{m}$, and 76.77ImVm$76.77I_{m}V_{m}$ compared to existing reconfiguration methodologies. [ABSTRACT FROM AUTHOR]

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form \[ L_A(X)=I_d{\otimes}I_n+A_1{\otimes}X_1+A_2{\otimes}X_2+\dots+A_g{\otimes}X_g\succeq 0, \]LA(X)=Id⊗In+A1⊗X1+A2⊗X2+⋯+Ag⊗Xg⪰0, where the $ A_i $ Ai and $ X_i $ Xi are symmetric matrices and the $ X_i $ Xi have any size $ n\times n $ n×n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme. [ABSTRACT FROM AUTHOR]

This paper investigates the spectral properties of block circulant matrices with high-order symmetric (or Hermitian) blocks. We analyze cases with dependent or sparse independent entries within these blocks. Additionally, we analyze the distribution of singular values for the product of independent circulant matrices with non-Hermitian blocks. [ABSTRACT FROM AUTHOR]

In this paper, we address the extremal inverse eigenvalue problem for pentadiagonal matrices. We provide sufficient conditions for their existence and realizability through new constructions that consider spectral data of its leading principal submatrices. Finally, we present some examples generated from the algorithmic procedures derived from our results. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability. [ABSTRACT FROM AUTHOR]

We examine a method for solving an infinite‐dimensional tensor eigenvalue problem Hx=λx$$ Hx=\lambda x $$, where the infinite‐dimensional symmetric matrix H$$ H $$ exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e−Ht$$ {e}^{- Ht} $$ is used to obtain an approximation to the desired eigenvector. This infinite‐dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step t$$ t $$ to ensure accurate and rapid convergence of the power method. [ABSTRACT FROM AUTHOR]

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method. [ABSTRACT FROM AUTHOR]

We propose and illustrate a general framework in which spatial autocorrelation is measured by the Frobenius product of two kernels, a feature kernel and a spatial kernel. The resulting autocorrelation index δ$$ \delta $$ generalizes Moran's index in the weighted, multivariate setting, where regions, differing in importance, are characterized by multivariate features. Spatial kernels can traditionally be obtained from a matrix of spatial weights, or directly from geographical distances. In the former case, the Markov transition matrix defined by row‐normalized spatial weights must be made compatible with the regional weights, as well as reversible. Equivalently, space is specified by a symmetric exchange matrix containing the joint probabilities to select a pair of regions. Four original weight‐compatible constructions, based upon the binary adjacency matrix, are presented and analyzed. Weighted multidimensional scaling on kernels yields a low‐dimensional visualization of both the feature and the spatial configurations. The expected values of the first four moments of δ$$ \delta $$ under the null hypothesis of absence of spatial autocorrelation can be exactly computed under a new approach, invariant orthogonal integration, thus permitting to test the significance of δ$$ \delta $$ beyond the normal approximation, which only involves its expectation and expected variance. Various illustrations are provided, investigating the spatial autocorrelation of political and social features among French departments. [ABSTRACT FROM AUTHOR]

We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin–Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties. [ABSTRACT FROM AUTHOR]

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Given a graph G with vertices { v 1 , ... , v n } , we define S (G) to be the set of symmetric matrices A = [ a i , j ] such that for i ≠ j we have a i , j ≠ 0 if and only if v i v j ∈ E (G). Motivated by the Graph Complement Conjecture, we say that a graph G is complementary vanishing if there exist matrices A ∈ S (G) and B ∈ S (G ‾) such that A B = O. We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in terms of certain minimal complementary vanishing graphs. In addition to this, we determine which graphs on at most 8 vertices are complementary vanishing. [ABSTRACT FROM AUTHOR]

Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related matrices, such as the adjacency matrix of an undirected, possibly weighted graph. [ABSTRACT FROM AUTHOR]

A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $ 0,1,\ldots,N $ 0 , 1 , ... , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented. [ABSTRACT FROM AUTHOR]

Symmetric non-negative matrix factorization (NMF) can naturally capture the embedded clustering structure in the graph representation. It is an important method for linear and nonlinear data clustering applications. However, it is sensitive to the initialization of variables, and the quality of the initialization matrix greatly affects the clustering performance. In semi-supervised clustering, it faces the challenge of learning a more discriminative representation from limited labeled data. This paper introduces a constrained propagation self-adaptive self-supervised non-negative matrix factorization clustering algorithm (CPS³NMF) to solve the above problems. The algorithm propagates finite constraints to unconstrained data points, constructing a similarity matrix imbued with constraint information. The resultant similarity matrix serves the role of a non-negative symmetric matrix decomposition in SNMF and functions as graph regularization for the assignment matrix, fully utilizing the limited constraint information to preserve the geometrical structure of data space. Concurrently, leveraging the sensitivity of initial features in SNMF, the algorithm employs adaptively learned weights to rank the quality of multiple initial matrices. By integrating results from multiple clustering attempts, it progressively enhances the performance of semi-supervised clustering. Experiments on 6 public datasets show that the proposed CPS³NMF algorithm outperforms other state-of-the-art algorithms, proving its effectiveness in semi-supervised clustering. [ABSTRACT FROM AUTHOR]

In 1992, Godsil and Hensel published a ground-breaking study of distance-regular antipodal covers of the complete graph that, among other things, introduced an important connection with equi-isoclinic subspaces. This connection seems to have been overlooked, as many of its immediate consequences have never been detailed in the literature. To correct this situation, we first describe how Godsil and Hensel's machine uses representation theory to construct equi-isoclinic tight fusion frames. Applying this machine to Mathon's construction produces q + 1 equi-isoclinic planes in R q + 1 for any even prime power q > 2. Despite being an application of the 30-year-old Godsil–Hensel result, infinitely many of these parameters have never been enunciated in the literature. Following ideas from Et-Taoui, we then investigate a fruitful interplay with complex symmetric conference matrices. [ABSTRACT FROM AUTHOR]

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph G π are a subset of its eigenvalues, i.e. σ (G π) ⊆ σ (G). A natural question is whether it is possible to recover the remaining eigenvalues σ (G) − σ (G π) in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to the removal of a single observation have been well documented in the literature. However, a complicating factor can exist in that the rank of the eigenvalues may change due to the removal of an observation, and with that so too does the perceived importance of the corresponding eigenvector. We refer to this problem as "switching of eigenvalues". Since there is not enough information in the new eigenvalues, post observation removal, to indicate that this has happened, how do we know that this switching has occurred? In this article, we show that approximations to the eigenvalues can be used to help determine when switching may have occurred. We then discuss possible actions researchers can take based on this knowledge, for example making better choices when it comes to deciding how many principal components should be retained and adjustments to approximate influence diagnostics that perform poorly when switching has occurred. Our results are easily applied to any eigenvalue problem involving symmetric matrix estimators. We highlight our approach with application to two real data examples. [ABSTRACT FROM AUTHOR]

Given a pair of real symmetric matrices A , B ∈ R n × n with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix C = [ A B I O ] ∈ R 2 n × 2 n. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C. [ABSTRACT FROM AUTHOR]

We study a variant of the Sherrington–Kirkpatrick (S–K) spin glass model with external field, where the random symmetric couplings matrix does not consist of i.i.d. entries but is instead orthogonally invariant in law. For sufficiently high temperature, we prove a replica-symmetric formula for the first-order limit of the model free energy. Our analysis is an adaptation of a conditional second-moment-method argument previously introduced by Bolthausen for studying the high-temperature regime of the S–K model, where one conditions on the iterates of an Approximate Message Passing (AMP) algorithm for solving the TAP equations for the model magnetization. We apply this method using a memory-free version of AMP that is tailored to the orthogonally invariant structure of the model couplings. [ABSTRACT FROM AUTHOR]

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit U q (gl N) -module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains. [ABSTRACT FROM AUTHOR]

We consider a wide class of closed convex cones K in the space of real n × n symmetric matrices and establish the existence of a chain of faces of K , the length of which is maximized at n (n + 1) 2 + 1. Examples of such cones include, but are not limited to, the completely positive and the copositive cones. Using this chain, we prove that the distance to polyhedrality of any closed convex cone K that is sandwiched between the completely positive cone and the doubly nonnegative cone of order n ≥ 2 , as well as its dual, is at least n (n + 1) 2 − 2 , which is also the worst-case scenario. [ABSTRACT FROM AUTHOR]

In this paper, we analytically derive closed-form expressions for the tangency portfolio weights: the fully invested portfolio that maximizes the expected return over the risk-free rate, relative to the volatility of the portfolio return. We explicitly derive this portfolio from a range of underlying return models and show examples where it coincides with different well-known smart beta products. Specifically, we find the closed-form expression for the tangency portfolio weights for a return model with compound symmetric correlation matrix. We also deduce the tangency portfolio weights for the CAPM return model and illustrate in a case study that the estimated tangency portfolio weights may distinctly deviate from the market value weighted portfolio. Furthermore, we show that depending on the return model, the tangency portfolio weights may take a diverse set of shapes; from very diversified to highly concentrated portfolios. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A P ≥ k -factor means a path-factor in which every component admits order at least k ( k ≥ 2 ). The distance matrix D (G) of G is an n × n real symmetric matrix whose (i, j)-entry is the distance between the vertices v i and v j . The distance signless Laplacian matrix Q (G) of G is defined by Q (G) = T r (G) + D (G) , where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η 1 (G) of Q (G) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a P ≥ 2 -factor in a graph and claim that the following statements are true: (i) G admits a P ≥ 2 -factor for n ≥ 4 and n ≠ 7 if η 1 (G) < θ (n) , where θ (n) is the largest root of the equation x 3 - (5 n - 3) x 2 + (8 n 2 - 23 n + 48) x - 4 n 3 + 22 n 2 - 74 n + 80 = 0 ; (ii) G admits a P ≥ 2 -factor for n = 7 if η 1 (G) < 25 + 161 2 . [ABSTRACT FROM AUTHOR]

In this paper, we present a statistical approach to evaluate the relationship between variables observed in a two-factors experiment. We consider a three-level model with covariance structure Σ ⊗ Ψ 1 ⊗ Ψ 2 , where Σ is an arbitrary positive definite covariance matrix, and Ψ 1 and Ψ 2 are both correlation matrices with a compound symmetric structure corresponding to two different factors. The Rao's score test is used to test the hypotheses that observations grouped by one or two factors are uncorrelated. We analyze a fermentation process to illustrate the results. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

• Heat conduction in perforated laminated plate by elliptical holes is solved using formulated special elements. • Special fundamental solutions are derived using Stroh formula and conformal mapping. • Special element containing a rotated elliptical hole is formulated in the concept of hybrid finite element. • Meshing effort around hole region is greatly alleviated and element boundary integration is featured. • The accuracy, efficiency and flexibility of the present special element are demonstrated. The high-efficient and accurate analysis of anisotropic composite media weakened by a large amount of holes or cuts is still a great challenge in practical composite engineering. In this work, specially-proposed polygonal finite element enclosing an arbitrarily oriented elliptical hole in general anisotropic composite media is formulated for thermal analysis in the framework of hybrid finite element theory. Two independent temperature field approximations are defined for the present special element: Intra-element temperature field and element frame temperature field. The former defined inside the element is written in the form of truncated summation of the weighted special fundamental solutions, which are derived using the Stroh complex theory and conformal mapping whilst the latter is implemented by the conventional shape function interpolation. Then the hybrid variational functional is employed to combine them together to generate an element stiffness equation with respect to nodal temperature and an optional relationship between the interpolation coefficients. The element stiffness matrix is symmetric and involves element boundary integrals only. Compared to the traditional finite element, the distinctive element-boundary-integral and enclosed-elliptical-hole features make the present element significantly improve the meshing effort around an elliptical hole by reducing the number of elements and decreasing the meshing difficulty, and more importantly, effectively capture the critical variation of physical fields around the elliptical hole without locally-refined meshes. Moreover, the present special element can be incorporated directly into the conventional finite element system without any difficulty for extensive analysis. Finally, several typical examples are simulated by the present special elements to demonstrate their effectiveness. [ABSTRACT FROM AUTHOR]

We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ R n × n with bounded entries (i.e., ‖ A ‖ ∞ ≤ 1 ). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ± ϵ n using those of a randomly sampled O ~ log 3 n ϵ 3 × O ~ log 3 n ϵ 3 principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of ± ϵ nnz (A) and ± ϵ ‖ A ‖ F when the rows of A can be sampled with probabilities proportional to their sparsities or their squared ℓ 2 norms respectively. Here nnz (A) is the number of non-zero entries in A and ‖ A ‖ F is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of A . We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice. [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce the readers, especially the researchers, to some topics connected with functions of matrix argument, scaling models, distributions of products and ratios, Bayesian structures, symmetric products and symmetric ratios of matrices, scalar and matrix-variate fractional integrals, functions of matrix argument through entropy optimization, singular matrix-variate gamma and beta functions etc which are currently active so that interested readers can get into these classes of problems for their current research or teaching. Let X be a p × q, p = q matrix of rank p in the real domain. If the function f(X), associated with X, is a function of XX', where a prime denotes the transpose, then such a function appears in a number of different disciplines. This paper examines the recent developments in such matrix-variate functions when X is in the real or complex domain. Connections to Bayes procedures, quantum physics, scalar and matrix texture models in communication and engineering problems, fractional integrals, distributions of symmetric products and symmetric ratios of matrices, singular matrix-variate gamma and beta functions and other related areas are pointed out. Only an overview of the current research in these topics with some illustrative examples are given in this paper. Since the material is summarized from the author's own works, most of the references are author's own papers, and hence similarity index, similarity with author's own works, may be high. The materials also cover some current results which are being published. [ABSTRACT FROM AUTHOR]

Let r and s be positive integers such that r ⩾ 2. Let U and V be vector spaces over fields F and K , respectively, such that dim ⁡ U ⩾ 3 and F has at least r + 1 elements. In this paper, we characterize surjective maps ψ : S r U → S s V preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim ⁡ U ⩾ 3 and the cardinality | F | ⩾ r + 1 in our result. [ABSTRACT FROM AUTHOR]

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about Ω and Φ, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices T-1n (Φ)Tn(Ω), given a monotonic ratio r=Ω/Φ. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for sum of secondary range symmetric matrices to be secondary range symmetric is given here. Also, the concept of parallel summable secondary range symmetric matrices and some of its characterizations are being explored. [ABSTRACT FROM AUTHOR]

In this paper, we contemplate addressing nonlinear problems involving complex symmetric Jacobian matrices. Firstly, we establish a parameter-free method called modified Newton–CAPRESB (MN–CAPRESB) method by harnessing the modified Newton method as the outer iteration and the CAPRESB (Chebyshev accelerated preconditioned square block) method as the inner iteration. Secondly, the local and semilocal convergence theorems of MN–CAPRESB method are proved under some conditions. Eventually, the numerical experiments of two kinds of complex nonlinear equations are presented to validate the feasibility of MN–CAPRESB method compared to other existing iteration methods. [ABSTRACT FROM AUTHOR]

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = il£) with L (or £) and U ( or il) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J^' = UL (or = £il), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations. [ABSTRACT FROM AUTHOR]

Recently, community detection has emerged as a prominent research area in the analysis of complex network structures. Community detection models based on non-negative matrix factorization (NMF) are shallow and fail to fully discover the internal structure of complex networks. Thus, this article introduces a novel constrained symmetric non-negative matrix factorization with deep autoencoders (CSDNMF) as a solution to this issue. The model possesses the following advantages: (1) By integrating a deep autoencoder to discern the latent attributes bridging the original network and community assignments, it adeptly captures hierarchical information. (2) Introducing a graph regularizer facilitates a thorough comprehension of the community structure inherent within the target network. (3) By integrating a symmetry regularizer, the model's capacity to learn undirected networks is augmented, thereby facilitating the precise detection of symmetry within the target network. The proposed CSDNMF model exhibits superior performance in community detection when compared to state-of-the-art models, as demonstrated by eight experimental results conducted on real-world networks. [ABSTRACT FROM AUTHOR]

The parameterized block splitting (PBS) is a convergent and efficient iterative method to solve the large complex symmetric linear systems. In this paper, by using PBS iterative technique, the Newton equation is approximately solved, then we establish the modified Newton-PBS iterative method to solve the complex nonlinear systems whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, the local convergence analysis are explored under appropriate conditions. Ultimately, we apply the new method and several known methods to experimental numerical examples, and experimental results verify the superiority and efficiency of our new method. Especially, in terms of CPU time and iteration steps, our method is obviously better. [ABSTRACT FROM AUTHOR]

In this paper, we advocate a new technique to determine inner bounds for the extreme eigenvalues of real symmetric matrices. Our method involves the matrix elements and compares favourably with existing methods. We also show how the bounds can be optimized. [ABSTRACT FROM AUTHOR]

The existence of real symmetric interval matrices is a natural phenomenon that can be found in a variety of real-life scenarios. In this particular feature, we develop the principal axis/spectral theorem for real symmetric interval matrices. This theorem paves the way for exciting developments in interval matrix theory, particularly concerning issues such as approximations, dealing with inexact or vague data, coping with the unavailability of data, and managing errors in measurements. To demonstrate the practical utility of our approach, we delve into a concrete real-world example involving vibration analysis within an uncertain environment. This research opens up new horizons for the effective utilization of interval matrices in addressing challenges arising from uncertainty and imprecision in diverse fields. [ABSTRACT FROM AUTHOR]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]