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For sequences α ≡ { α n } n = 0 ∞ of positive real numbers, called weights, we study the weighted shift operators W α having the property of moment infinite divisibility ( MID ); that is, for any p > 0 , the Schur power W α p is subnormal. We first prove that W α is MID if and only if certain infinite matrices log ⁡ M γ ( 0 ) and log ⁡ M γ ( 1 ) are conditionally positive definite (CPD). Here γ is the sequence of moments associated with α, M γ ( 0 ) , M γ ( 1 ) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of W α , and log is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between k–hyponormality and n–contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift W α is MID if and only if for all p > 0 , M γ p ( 0 ) and M γ p ( 1 ) are CPD.

Measurement error models are studied extensively in the literature. In these models, when the measurement error variance $$\varvec{\Sigma _{\delta \delta }}$$ is known, the estimating techniques require positive definiteness of the matrix $$\varvec{S_{xx}-\Sigma _{\delta \delta }}$$ , even when this is positive definite, it might be near singular if the number of observations is small. There are alternative estimators discussed in literature when this matrix is not positive definite. In this paper, estimators when the matrix $$\varvec{S_{xx}-\Sigma _{\delta \delta }}$$ is near singular are proposed and it is shown that these estimators are consistent and have the same asymptotic properties as the earlier ones. In addition, we show that our estimators work far better than the earlier estimators in case of small samples and equally good for large samples.

The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.

An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.

This study associates the symmetric Bernoulli matrix with various combinatorial matrices, such as Pascal, Vandermonde, and Stirling matrices of the first and second kind including Catalan numbers by way of dual matrices of the matrix. A recurrence relation related to the entries of the Cholesky factor of a symmetric positive definite matrix is presented, which sheds new light on examining positive definiteness and Cholesky's method of a symmetric matrix. This relation is applied to give a necessary and sufficient condition for the positive definiteness of a symmetric matrix, which leads to a slightly modified form of the outer-product formulation for Cholesky's method of a symmetric matrix.

We introduce the concept of angular correlation for estimating the instantaneous correlation matrix with a single multivariate realization. The proposed estimator is generally a positive definite correlation matrix and robust in that for bivariate normal data, the sample angular correlation is equally likely to be above or below the population correlation coefficient. We then generalize the dynamic conditional correlation (DCC) model to the dynamic conditional angular correlation (DCAC) model. We demonstrate the efficacy and robustness of the proposed methods against leptokurticity, with some numerical experiments. In particular, a real application illustrates the better performance of the DCAC model than the DCC model in portfolio construction.

In this paper, we introduce interval tensors and present some results about their eigenvalues, positive definiteness and application in solving multi-linear systems. It is proved that the set of maximum Z-eigenvalues of a symmetric interval tensor is a compact interval. Also, several bounds for eigenvalues of an interval tensor are proposed. In addition, necessary and sufficient conditions for having a positive definite interval tensor are presented and investigated. Furthermore, solving tensor equations using interval methods is presented and the interval Jacobi and Gauss–Seidel algorithms are extended for interval multi-linear systems. Finally, some numerical experiments are carried out to illustrate the methods.

The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, predictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.

The article deals with the possibility of using the square-root filtering methods to guarantee numerical stability of continuous identification during long-term monitoring of time-varying industrial systems. The main emphasis is being placed on long-term deployment, therefore the process of recursive computation of the covariance matrix $P_{(k)}$ in the form of regression analysis is discussed in detail. The application of regression analysis in form of a generalized least-squares method for the purpose of weighting a posteriori information is solved by using the square-root filtering methods, since the methods guarantee, that the difference between two positive definite matrices and the result are positive definite matrices. Also, the square-root filtering suppresses the negative effect of errors caused by insumcient length of computers bit word, rounding, etc., which leads to the loss of numerical stability of the algorithm and also to the loss of positive definiteness of the matrices. The described methodology is being used to create continuous identification forgetting and weighting algorithms. These algorithms are used for long-term identification and creation of mathematical models of the monitored industrial systems. The obtained models are then being used for various applications, mostly for applications which require predictive control.

In this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$ satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.

The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized.

Although isotropy, symmetry and separability are commonly assumed for practical reasons, anisotropic, asymmetric and non-separable covariance functions are often more realistic; in addition, strict positive definiteness is also desirable, since it ensures the invertibility of the kriging coefficient matrix. In this paper, a critical review of these concepts is proposed and it is shown how these aspects are strictly related. In particular, separable covariance models represent a simple way to construct component-wise anisotropic models which, under suitable conditions, are strictly positive definite. Similarly, some other results on strict positive definiteness can be used to obtain non-separable anisotropic models. Covariance functions defined on partially overlapped domains are used to construct non-geometric spatial anisotropic covariance functions, also characterized by non-separability and strict positive definiteness. Moreover, anisotropic and asymmetric covariance functions that are also strictly positive definite are presented.

As an extension of linear complementary problem, tensor complementary problem has been effectively applied in \begin{document}$ n $\end{document} -person noncooperative game. And a multitude of researchers have focused on its properties and theories, while the valid algorithms for tensor complementary problem is still deficient. In this paper, stimulated by the potential reduction method for linear complementarity problem, we present a new algorithm for the tensor complementarity problem, which combines the idea of damped Newton method and the interior point method. Utilizing the new algorithm, we settle the tensor complementary problem with the underlying tensor being diagonalizable and positive definite. Furthermore, the global convergence of the iterative scheme is theoretically guaranteed and the given preliminary numerical experiments indicate the efficiency of the method.

We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under \emph{positive-definite} assumption on the disorder covariance matrix $\Delta^2$. First, we prove exponential overlap concentration at high temperature for both \indf~and \emph{positive-definite} $\Delta^2$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupling argument to prove overlap concentration up to $\beta_c$, which is expected to be the critical inverse temperature. The argument holds for both \emph{positive-definite} and emph{indefinite} $\Delta^2$, and $\beta_c$ has the same expression in two different cases. Second, we develop a species-wise cavity approach to study the overlap fluctuation, and the asymptotic variance-covariance matrix of overlap is obtained as the solution to a matrix-valued linear system. The asymptotic variance also suggests the de Almeida--Thouless (AT) line condition from the Replica Symmetry (RS) side. Our species-wise cavity approach does not require the positive-definiteness of $\Delta^2$. However, it seems that the AT line conditions in \emph{positive-definite} and \emph{indefinite} cases are different. Finally, in the case of \emph{positive-definite} $\Delta^2$, we prove that above the AT line, the MSK model is in Replica Symmetry Breaking phase under some natural assumption. This generalizes the results of J.~Stat.~Phys.~174 (2019), no.~2, 333--350, from 2-species to general species., Comment: 35 pages, 1 figure. Final version. To appear in J.~Stat.~Phys

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property. More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straightforward. H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix B allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B = B − ⊕ B 0 ⊕ B + . The interesting part is B 0 , as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B − , B 0 , and B + . Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint. For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B 0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B 0 is used.

This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes.\ Under the framework of quasi-metric spaces and isometric embeddings, the paper proposes a general and unifying framework that encompasses results provided by earlier literature.\ Our results allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or high dimensional spheres and quasi-metric spaces.\ In turn, Gneiting's theorem is proved here by a direct construction, eluding Fourier inversion (the so-called Gneiting's lemma) and convergence arguments that are required by Gneiting to preserve an integrability assumption.

Axial symmetry for covariance functions defined over spheres has been a very popular assumption for climate, atmospheric, and environmental modeling. For Gaussian random fields defined over spheres embedded in a three-dimensional Euclidean space, maximum likelihood estimation techiques as well kriging interpolation rely on the inverse of the covariance matrix. For any collection of points where data are observed, the covariance matrix is determined through the realizations of the covariance function associated with the underlying Gaussian random field. If the covariance function is not strictly positive definite, then the associated covariance matrix might be singular. We provide conditions for strict positive definiteness of any axially symmetric covariance function. Furthermore, we find conditions for reducibility of an axially symmetric covariance function into a geodesically isotropic covariance. Finally, we provide conditions that legitimate Fourier inversion in the series expansion associated with an axially symmetric covariance function.

This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.

Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. In this work we investigate different methods to generate random symmetric positive definite matrices with undirected graphical constraints. We show that if the graph is chordal it is possible to sample uniformly from the set of correlation matrices compatible with the graph, while for general undirected graphs we rely on a partial orthogonalization method., Improved figures, algorithm descriptions and text exposition. arXiv admin note: substantial text overlap with arXiv:1807.03090

In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors. We first show that the concerned (strongly) paired symmetric tensor is positive definite if and only if its smallest M -eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M -eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, by which we can check positive definiteness of the concerned tensor. The preliminary numerical results confirm our theoretical findings.

We consider approximation of a given positive definite matrix by nonnegative definite banded Toeplitz matrices. We show that the projection on linear space of Toeplitz matrices does not always preserve nonnegative definiteness. Therefore we characterize a convex cone of nonnegative definite banded Toeplitz matrices which depends on the matrix dimensions, and we show that the condition of positive definiteness given by Parter [{\em Numer. Math. 4}, 293--295, 1962] characterizes the asymptotic cone. In this paper we give methodology and numerical algorithm of the projection basing on the properties of a cone of nonnegative definite Toeplitz matrices. This problem can be applied in statistics, for example in the estimation of unknown covariance structures under the multi-level multivariate models, where positive definiteness is required. We conduct simulation studies to compare statistical properties of the estimators obtained by projection on the cone with a given matrix dimension and on the asymptotic cone.

Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e., $${f_{\alpha ,c}}\left( { - x} \right)\overline {{f_{\alpha ,c}}\left( x \right)} ,x \in \mathbb{R};$$ , x ∈ R; (2) fα,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function fα,c is linear and satisfies the conditions fα,c(0) = 1, fα,c(α) = c, and fα,c(1) = 0. It is proved that the complex piecewise linear function fα,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), $$where m\left( \alpha \right) = \left\{ \begin{gathered} 0if1/\alpha \notin \mathbb{N}, \hfill \\ - \alpha if1/\alpha \in \mathbb{N}. \hfill \\ \end{gathered} \right.$$ If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.

Quasi-Newton methods based on the symmetric rank-one (SR1) update have been known to be fast and provide better approximations of the true Hessian than popular rank-two approaches, but these properties are guaranteed under certain conditions which frequently do not hold. Additionally, SR1 is plagued by the lack of guarantee of positive definiteness for the Hessian estimate. In this paper, we propose cubic regularization as a remedy to relax the conditions on the proofs of convergence for both speed and accuracy and to provide a positive definite approximation at each step. We show that the n-step convergence property for strictly convex quadratic programs is retained by the proposed approach. Extensive numerical results on unconstrained problems from the CUTEr test set are provided to demonstrate the computational efficiency and robustness of the approach.

The Markowitz critical line method for mean–variance portfolio construction has remained highly influential today, since its introduction to the finance world six decades ago. The Markowitz algorithm is so versatile and computationally efficient that it can accommodate any number of linear constraints in addition to full allocations of investment funds and disallowance of short sales. For the Markowitz algorithm to work, the covariance matrix of returns, which is positive semi-definite, need not be positive definite. As a positive semi-definite matrix may not be invertible, it is intriguing that the Markowitz algorithm always works, although matrix inversion is required in each step of the iterative procedure involved. By examining some relevant algebraic features in the Markowitz algorithm, this paper is able to identify and explain intuitively the consequences of relaxing the positive definiteness requirement, as well as drawing some implications from the perspective of portfolio diversification. For the examination, the sample covariance matrix is based on insufficient return observations and is thus positive semi-definite but not positive definite. The results of the examination can facilitate a better understanding of the inner workings of the highly sophisticated Markowitz approach by the many investors who use it as a tool to assist portfolio decisions and by the many students who are introduced pedagogically to its special cases.

Given a symmetric positive definite (s.p.d.) matrix, two methods are proposed for directly constructing hierarchically semiseparable (HSS) matrix approximations that are guaranteed to be positive d...

It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-\theta)^\delta P_n^{(\alpha-\frac12,\beta-\frac12)}(\cos \theta) \left(\sin \tfrac{\theta}2\right)^{2 \alpha} \left(\cos \tfrac{\theta}2\right)^{2 \beta} d\theta > 0 \end{equation*} for all $t \in (0,\pi]$ and $n \in \mathbb{N}$ if $\delta \ge \alpha + 1$ for $\alpha,\beta \in \mathbb{N}_0$ and $\max\{\alpha,\beta\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that implies the strictly positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ on the unit sphere $\mathbb{S}^{d-1}$ for $\delta \ge \lceil \frac{d}{2}\rceil$ and the Poly\`a criterion for positive definite functions on the sphere for all dimensions. Moreover, the positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ is also established on the compact two-point homogeneous spaces., Comment: Final form. Accepted Proc. AMS

Let α ∈ ( 0 , 1 ) , h ∈ R and let f α , h be an even function with the properties: f α , h ( x ) = 0 for x ∈ ( 1 , + ∞ ) , f α , h is linear over the intervals [ 0 , α ] and [ α , 1 ] , f α , h ( 0 ) = 1 , f α , h ( α ) = h , and f α , h ( 1 ) = 0 . In this paper we prove that f α , h is positive definite on R ⟺ m ( α ) ≤ h ≤ 1 − α , where m ( α ) = 0 if 1 / α ∉ N , and m ( α ) = − α if 1 / α ∈ N .

For a broad class of functions f: [0,+∞) → ℝ, we prove that the function f(ρ λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ≢ 0 and α(E, ρ) is the Schoenberg constant.

In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem., 21 pages, 1 figure. Revised version, minor changes and references added

We consider the functions hμ,ν introduced by Zastavnyi in 2002. The family of these functions is a subfamily of Buhmann’s functions and contains the families of functions introduced by Trigub in 1987 and Wendland in 1995. We investigate the problems of positive definiteness and smoothness at zero for the linear combinations β2ehμ,ν (x/β2) − β1eμ,ν (x/β1).

General necessary conditions on the real parameters α, β, C, D for the function $${e^{ - \alpha \rho \left( x \right)}}\left( {C\cos \beta \rho \left( x \right) + D\sin \beta \rho \left( x \right)} \right),$$ where ρ is the norm on R n , to be positive definite on R n , are obtained. For ρ(x) = ‖x‖ p , a criterion on these parameters is obtained in the following cases: (i) p = 1or p = 2; (ii) 3 < p ≤ ∞ and n = 2.

The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.

The positive (semi-)definiteness of even-order tensors has been widely studied in these years due to its applications in various aspects, such as spectral hypergraph theory, automatic control, polynomial theory, stochastic process, magnetic resonance imaging and so on. It has been shown that $$\mathcal {M}$$ -tensors, $$\mathcal {B}$$ -tensors, $$\mathcal {H}$$ -tensors, Hilbert tensors and stochastic tensors can be positive definite under proper conditions. However, there are still many positive definite tensors that can not be determined by the above criteria. In this paper, we provide a new class of positive definite tensors whose non-diagonal entries can be positive compared to (strong) $$\mathcal {M}$$ -tensors, and we call it strong $$\mathcal {M}_z$$ -tensors, which can arise from discretizing differential equations, since it is based on Z-eigenvalues rather than H-eigenvalues traditionally. Moreover, we show that an even-order (strong) $$\mathcal {M}$$ -tensor must be an (a strong) $$\mathcal {M}_z$$ -tensor, which reflects the inclusion relationship between even-order $$\mathcal {M}$$ -tensors and $$\mathcal {M}_z$$ -tensors. We also introduce (strong) $$\mathcal {H}_z$$ -tensors, as a generalization of (strong) $$\mathcal {M}_z$$ -tensors, and its positive semi-definiteness (positive definiteness) has been studied. Finally, some conditions are given for a tensor to be an (a strong) $$\mathcal {M}_z$$ -tensor and we use it to study the stability of a high-order nonlinear system.

In this paper, we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion., 26 pages

In [7] a conjecture relating the positive definiteness of a similarity with its transitivity with respect to the Lukasiewicz t-norm is made. In its current form, the conjecture is not true but from a modified version interesting consequences can be derived.

The main contribution in the present paper is a characterization for positive definiteness and strict positive definiteness of a kernel on the product $$X \times S^d$$ , in which X is a nonempty set and $$S^d$$ is the usual d-dimensional unit sphere in Euclidean space, through Fourier-like expansions. The setting presupposes continuity and isotropy on the $$S^d$$ side and no algebraic structure or topology on X. The result may be interpreted as another extension of a classical result of I. J. Schoenberg on positive definite functions on spheres. We take a closer look at our results in the case in which X is a locally compact group, paying special attention to usual Euclidean spaces and high dimensional tori.

The Quasi-Newton methods do not compute the Hessian of nonlinear functions. The Hessian is updated by analyzing successive gradient vectors instead. The Quasi-Newton algorithm was first proposed by William C. Davidon, a physicist while working at Argonne National Laboratory, United States in 1959. In Newton’s method, we require to compute the inverse of the Hessian at every iterations which is a very expensive computation. It requires an order n cube effort if n is the size of the Hessian. These drawbacks of Newton’s method gave motivation to develop the Quasi-Newton methods. The idea of Quasi-Newton method is that to approximate the inverse of Hessian by some other matrix which should be positive definite so that we can get a good approximation of the Hessian inverse at a given iteration. This saves the work of computation of second derivatives and also avoids the difficulties associated with the loss of positive definiteness. This approximate matrix is updated on every iteration so that as the search direction proceeds the second-order derivative information improves. In this chapter, we present three important Quasi-Newton methods, and they are Rank One Correction Formula method, Davidon Fletcher Powell method, and Broyden–Fletcher–Goldfarb–Shanno method.

Necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix $A(u)$ are derived, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since $A(u)$ is typically neither symmetric nor positive definite. In particular, the normal ellipticity of $A(u)$ for all $u$ and the symmetry of the Onsager matrix implies its positive definiteness and hence an entropy structure. If $A$ is constant or nearly constant in a certain sense, the existence of an entropy structure is equivalent to the normal ellipticity of $A$. Several applications and examples are presented, including the $n$-species population model of Shigesada, Kawasaki, and Teramoto, a volume-filling model, and a fluid mixture model with partial pressure gradients. Furthermore, the normal elipticity of these models is investigated and some extensions are discussed.

We investigate the notion of conditionally positive definite in the context of Hilbert $$C^*$$ -modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert $$C^*$$ -modules. As a consequence, we show that a $$C^*$$ -metric space (S, d) is $$C^*$$ -isometric to a subset of a Hilbert $$C^*$$ -module if and only if $$K(s,t)=-d(s,t)^2$$ is a conditionally positive definite kernel. We also present a characterization of the order $$K'\le K$$ between conditionally positive definite kernels.

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, covering almost all possible...

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the torus.

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.

Some new criteria for identifying ℋ-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J.Matrix Anal. Appl., 2010, 31: 2517–2530]. Numerical examples are reported to demonstrate the corresponding results.

Identifying (conditionally) strictly positive definite functions is of great importance as they allow the unique solution of certain interpolation problems. We introduce new sufficient (and some necessary) conditions for functions to be conditionally strictly positive definite on all spheres S d − 1 , d > 2 , only employing monotonicity properties. For strictly positive definite and conditionally negative definite functions on all spheres we give a characterisation in terms of monotonicity properties. Further, we prove that multiply monotone functions are positive definite on spheres up to a certain dimension.

In this article, we focus on the connections of monotonicity properties and the strict positive definiteness of functions on R d . We collect the existing results known for the Euclidean space and present a new technique to construct positive definite functions from multiply monotone functions. Further, we collect properties of multiply monotone functions which allow us to construct new positive definite functions.

Identifying the positive definiteness of an even-degree homogeneous polynomial form arises in various applications, such as evolutionary game dynamics, automatical control, magnetic resonance imaging, and spectral hypergraph theory. However, it is difficult to determine whether a given homogeneous polynomial is positive definite or not because the problem is NP-hard. In this paper, an iterative algorithm of identifying the positive definiteness of irreducible homogeneous polynomial forms is proposed by identifying strong H -tensors with weakly irreducibility. The validity of the iterative algorithm is proved theoretically. Experimental results on multilinear systems, high-order Markov chains and symmetric multi-player games are presented to illustrate the applications of the proposed methods.

Positive definite functions are fundamental to many areas of applied mathematics, probability theory, spatial statistics and machine learning, amogst others. Motivated by a problem coming from the maximum likelihood estimation under fixed domain asymptotics, we consider a new operator acting on rescaled weighted differences between two members of the class $\Phi_d$ of positive definite radial functions,. In particular, we study the positive definiteness of the operator for the Mat\'ern, Generalized Cauchy and Generalized Wendland families. It turns out that proposed operator allows to govern differentiability at the origin, and to attain negative correlations., Comment: 12 pages, 3 figures