Search

In the year 1982, John Chollet conjectured that, for any pair of $ n\times n $ n × n positive semidefinite matrices A, B, $ per(A)\cdot per(B)\geq per(A\circ B) $ per (A) ⋅ per (B) ≥ per (A ∘ B) , where $ A\circ B $ A ∘ B is the Hadamard product of A and B. This conjecture was proved to be valid for n = 2, 3 in the year 1987. In this paper, we show that the conjecture holds true for n = 4. [ABSTRACT FROM AUTHOR]

Catalan–Stieltjes matrices and their corresponding Catalan-like numbers are very important in combinatorics. In this paper, we give a sufficient condition for the Catalan–Stieltjes matrix to be triangular strictly totally positive (Δ STP), that is, all its minors are positive except those which are zero due to the triangular structure. On this basis, we show that it is possible to compute the bidiagonal factorization of a class of Catalan–Stieltjes matrices with high relative accuracy (HRA). We also demonstrate that this class of matrices includes many well-known combinatorial triangles, such as the Catalan triangle of Aigner and the Bell triangle. Moreover, we construct an accurate algorithm to compute the bidiagonal factorization of some Catalan–Stieltjes matrices. The algorithm guarantees HRA for several linear algebraic computations associated with these matrices, like computing their singular values, their inverses, as well as the solutions of some linear systems. We show the accuracy and effectiveness of the proposed algorithm through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider the notions of spherical p, log , and w-hyponormality of commuting pairs of operators and study their properties and relations. We first show that spherically p-hyponormal 2-variable weighted shifts of hyponormal operators are invariant under powers and the generalized spherical Aluthge transformation. Second, we study the relation between spherical p-hyponormality and spherical w-hyponormality. Third, we study whether the spherical p-hyponormality with 0

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces $ \ell ^p(I) , $ ℓ p (I) , where I is a nonempty set and $ p\in [1,+\infty). $ p ∈ [ 1 , + ∞). We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices S n that leave invariant the closed convex cones of copositive and completely positive matrices (COPn and CPn). A description of an invertible linear map on S n such that L(CPn) ⊂ CPn is obtained in terms of semipositive maps over the positive semidefinite cone S + n and the cone of symmetric nonnegative matrices N + n for n ⩽ 4, with specific calculations for n = 2. Preserver properties of the Lyapunov map X ↦ AX + XAt, the generalized Lyapunov map X ↦ AXB + BtXAt, and the structure of the dual of the cone π(CPn) (for n ⩽ 4) are brought out. We also highlight a different way to determine the structure of an invertible linear map on S 2 that leaves invariant the closed convex cone S + 2 . [ABSTRACT FROM AUTHOR]

For positive real numbers r , p 0 , and p 1 < ⋯ < p n , let K r be the Kraus matrix whose (i, j) entry is equal to 1 p i - p j ( p i r - p 0 r p i - p 0 - p j r - p 0 r p j - p 0 ). In this article, we give a supplemental result to Sano and Takeuchi (J. Spectr. Theory, 2022) about the Kraus matrices K r : the simplicity of non-zero eigenvalues. Our proof is accomplished by arguments similar to those for Loewner matrices given by Bhatia, Friedland and Jain (Indiana Univ. Math. J., 2016). [ABSTRACT FROM AUTHOR]

Let A be a positive semidefinite matrix. It is known that the Hadamard exponential of A is positive semidefinite; it is positive definite if and only if no two columns of A are identical. We give an alternative proof of the latter part with an application to Hadamard inverses. [ABSTRACT FROM AUTHOR]

We briefly review some history and some of the main mathematical properties of the basic reproduction number, a concept encountered frequently in mathematical biology. We discuss some limitations on the practical use of basic reproduction numbers. One stems from the known fact that basic reproduction numbers are not unique. Another is related to the fact that basic reproduction numbers and the spectral radius of their associated linear operator could respond in opposite ways when controlled. [ABSTRACT FROM AUTHOR]

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J -total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the J -totally nonnegative flag variety has a cellular decomposition into totally positive J -Richardson varieties. Moreover, each totally positive J -Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J -Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J -total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U − for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups. [ABSTRACT FROM AUTHOR]

Several spectral radii formulas for infinite bounded non-negative matrices in max algebra are obtained. We also prove some Perron–Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are similar to results on Frobenius normal form of $ n \times n $ n × n matrices. Some continuity results are also established. [ABSTRACT FROM AUTHOR]

We improve the generalized numerical radius version of the Heinz means inequalities by employing convexity and the Hermit-Hadamard inequality. Moreover, we present novel proofs for the generalized numerical radius of Heinz-type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the limit, unique solution of the nonlinear equations, geodesic property, tolerance relations and pinch on the spectral geometric mean for two positive definite operators. We show that the spectral geometric mean is a geodesic with respect to some semi-metric. We also prove that the tolerance relation on determinant one matrices can be characterized by the spectral geometric mean. Moreover, two positive tuples can be pinched by the spectral geometric mean. [ABSTRACT FROM AUTHOR]

Using the connection between ellipsoids and positive semidefinite matrices we provide alternative proofs to some recently proven inequalities concerning the volume of L 2 zonoids as consequences of classical inequalities for matrices. [ABSTRACT FROM AUTHOR]

Inspired by the idea and technique in a recent paper of Huy et al. (Linear Algebra Appl 656: 368–384, 2023), Ighachane and Bouchangour showed an important refinement of the results by Sababheh (Linear Algebra Appl 506: 588–602, 2016) involving both convex and log-convex functions. In this paper, we continue to establish further new inequalities for both convex and log-convex functions with some applications to scalar and matrix means as well as matrix norm and numerical radius. The significance of the obtained results is two folded: the results themselves and the way they extend many known results in the literature. [ABSTRACT FROM AUTHOR]

We study the cone of completely positive (cp) matrices for the first interesting case n = 5 . This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem. [ABSTRACT FROM AUTHOR]

We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia (n+,n−,nz,2np)\left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n+≥1{n}_{+}\ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n+≥1{n}_{+}\ge 1 and n−≥2{n}_{-}\ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order nn, we show that as n→∞n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for nn odd and 3/4 for nn even.

A complete flag in R n is a sequence of nested subspaces V 1 ⊂ ⋯ ⊂ V n - 1 such that each V k has dimension k. It is called totally nonnegative if all its Plücker coordinates are nonnegative. We may view each V k as a subspace of polynomials in R [ x ] of degree at most n - 1 , by associating a vector (a 1 , ⋯ , a n) in R n to the polynomial a 1 + a 2 x + ⋯ + a n x n - 1 . We show that a complete flag is totally nonnegative if and only if each of its Wronskian polynomials Wr (V k) is nonzero on the interval (0 , ∞) . In the language of Chebyshev systems, this means that the flag forms a Markov system or ECT-system on (0 , ∞) . This gives a new characterization and membership test for the totally nonnegative flag variety. Similarly, we show that a complete flag is totally positive if and only if each Wr (V k) is nonzero on [ 0 , ∞ ] . We use these results to show that a conjecture of Eremenko (Arnold Math J 1(3):339–342, 2015) in real Schubert calculus is equivalent to the following conjecture: if V is a finite-dimensional subspace of polynomials such that all complex zeros of Wr (V) lie in the interval (- ∞ , 0) , then all Plücker coordinates of V are real and positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko. (J Am Math Soc 22(4):909–940, 2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive. We also show that our conjecture is equivalent to a totally positive version of the secant conjecture of Sottile (2003). [ABSTRACT FROM AUTHOR]

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between $ (B^{ts/2}A^{(1-t)s} B^{ts/2})^{1/s} $ (B ts / 2 A (1 − t) s B ts / 2 ) 1 / s and the t-spectral mean $ A\natural_t B :=(A^{-1}\sharp B)^{t}A(A^{-1}\sharp B)^{t} $ A ♮ t B := (A − 1 ♯B) t A (A − 1 ♯B) t of two positive semidefinite matrices A and B, where $ A\sharp B $ A ♯B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature. [ABSTRACT FROM AUTHOR]

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as \[ P_\mu(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right)^{1/p} A^{1/2}\quad (p \ge 0). \] P μ (p , A , B) = A 1 / 2 (1 + (A − 1 / 2 B A − 1 / 2 ) p 2) 1 / p A 1 / 2 (p ≥ 0). In this paper, for $ 0\le p \le 1 \le q $ 0 ≤ p ≤ 1 ≤ q , we show that if one of the following inequalities \[ f(P_\mu(p, A, B)) \le f(P_\mu(1, A, B)) \le f(P_\mu(q, A, B)) \] f (P μ (p , A , B)) ≤ f (P μ (1 , A , B)) ≤ f (P μ (q , A , B)) holds for any positive definite matrices A and B, then the function f is operator monotone on $ (0, \infty). $ (0 , ∞). We also study the inverse problem for non-Kubo-Ando matrix power means with the powers 1/2 and 2. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means. [ABSTRACT FROM AUTHOR]

Let A, B be any two positive definite n × n matrices and Y be any n × n matrix. We aim to study the precised estimates of the eigenvalues of M Y (A , B) = A A 1 2 Y B 1 2 B 1 2 Y ⋆ A 1 2 B , with different choices of Y as contractive, expansive or unitary matrix respectively. As applications, we also discuss the block matrices involving several means of the matrices as their entries. [ABSTRACT FROM AUTHOR]

This study devotes to Heisenberg’s uncertainty inequalities of complex-valued functions in two free metaplectic transformation (FMT) domains without the assumption of orthogonality. In our latest work (Zhang in J Fourier Anal Appl 27(4):68, 2021), it is crucial that the FMT needs to be orthogonal for a decoupling of the cross terms. Instead of applying the orthogonality assumption, our current work uses the trace inequality for the product of symmetric matrices and positive semidefinite matrices to address the problem of coupling between cross terms. It formulates two types of lower bounds on the uncertainty product of complex-valued functions for two FMTs. The first one relies on the minimum eigenvalues of A j T A j - B j T A j , B j T A j , B j T B j - B j T A j , while the other one relies on the minimum eigenvalues of A j T A j + B j T A j , B j T B j + B j T A j and the maximum eigenvalues of B j T A j , where A j , B j , j = 1 , 2 are the blocks found in symplectic matrices. Also, they are all relying on the covariance and absolute covariance. Sufficient conditions that truly give rise to the lower bounds are obtained. The theoretical results are verified by examples and experiments. [ABSTRACT FROM AUTHOR]

We have recently introduced and studied the concept of singular value of a rectangular matrix relative to a pair of closed convex cones. Such cones act as constraint sets for the left-singular and right-singular vectors. This work extends the theory of cone-constrained singular value problems from rectangular matrices to linear maps between Euclidean spaces. [ABSTRACT FROM AUTHOR]

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of non-negative matrices that define operators on Banach sequences, we obtain additional results. Our results extend the results of several authors that appeared relatively recently. [ABSTRACT FROM AUTHOR]

By properties of Cvetković-Kostić-Varga-type (or, for short, CKV-type) B-matrices, a new class of nonsingular matrices called CKV-type B ¯ -matrices is given, and a new inclusion interval of the real eigenvalues of real matrices is presented. It is shown that the new inclusion interval is sharper than those provided by J. M. Peña (2003), and by H. B. Li et al. (2007). We also propose a direct algorithm for computing the new inclusion interval. Numerical examples are included to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by Schürmann (in: Quantum probability and applications II, proceedings of a 2nd workshop, Heidelberg/Germany 1984, lecture notes in mathematics, vol 1136, pp 475–492, 1985). It characterizes the generators of semigroups of linear maps on M n (C) which are k-positive, k-superpositive, or k-entanglement breaking. As a corollary we reprove Lindblad, Gorini, Kossakowski, Sudarshan’s theorem (J Math Phys 17:821, 1976; Commun Math Phys 48:119-130, 1976). We present some concrete examples of semigroups of operators and study how their positivity properties can improve with time. [ABSTRACT FROM AUTHOR]

Let A be a real matrix of size m × n . In classical linear algebra, a real number σ is called a singular value of A if there exist unit vectors u ∈ R m and v ∈ R n such that A v = σ u and A ⊤ u = σ v . In variational analysis, a singular value of A is viewed as a critical value of the bilinear form ⟨ u , A v ⟩ when u and v range on the unit spheres of R m and R n , respectively. If u and v are further required to be nonnegative, then the idea of criticality is expressed by means of a pair of complementarity problems, namely, 0 ≤ u ⊥ (A v - σ u) ≥ 0 and 0 ≤ v ⊥ (A ⊤ u - σ v) ≥ 0 . The parameter σ is now called a Pareto singular value of A. In this work we study the concept of Pareto singular value and, by way of application, we analyze a problem of nonnegative matrix factorization. The set Ξ (A) of Pareto singular values of A is nonempty and finite. We derive an explicit formula for the maximum number of Pareto singular values in a matrix of prescribed size. The elements of Ξ (A) can be found by solving a collection of classical singular value problems involving the principal submatrices of A. Unfortunately, such a method is cost prohibitive if m and n are large. For matrices of large size we propose an algorithm of alternating minimization type. This work is a continuation of our paper entitled Cone-constrained singular value problems published in the Journal of Convex Analysis (30, 2023, pp. 1285–1306). [ABSTRACT FROM AUTHOR]

In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose the conformal energy minimization (CEM), stretching energy minimization (SEM) and volume stretching energy minimization (VSEM) algorithms by using the Jacobi conformal projection to compute the ellipsoidal conformal, area- and volume-preserving parameterizations from the boundary of a simply connected closed 3-manifold M to the surface of an ellipsoid E 3 (a , b , c) and from the 3-manifold M to an ellipsoid E 3 (a , b , c) , respectively. At each correction step of SEM and VSEM, the coefficients of the Laplacian matrices are modified by imposing local area/volume stretch factors in the denominators. Furthermore, to find the area-preserving optimal mass transportation (OMT) map between ∂ M and ∂ E 3 (a , b , c) and the volume-preserving OMT map between M and E 3 (a , b , c) , in light of SEM and VSEM, we propose the ellipsoidal area-preserving OMT (AOMT) and volume-preserving OMT (VOMT) algorithms, which are combined with the project gradient method, while preserving the local area/volume ratios and minimizing the transport costs and distortions. The numerical results demonstrate that the transformation of a 3D irregular image into an appropriate ellipsoid or cuboid incurs a smaller transport cost and reduces the difference in the conversion compared with that into a ball or cube. [ABSTRACT FROM AUTHOR]

We use a family of atomic decomposition operators to determine an explicit function realising the complex interpolation between two mixed norm weighted Bergman spaces in tube domains over open convex homogneous cones. Our results extend and improve earlier work from (Békollé et al. in C R Acad Sci Paris Ser I(337):13–18, 2003), where the problem was considered for scalar powers α = (α , … , α) and symmetric cones Ω . [ABSTRACT FROM AUTHOR]

In this paper, we study the practical exponential stability of nonlinear nonautonomous differential equations under nonlinear perturbations. By introducing a new method, we obtain some explicit criteria for the practical exponential stability of these equations. Furthermore, several characterizations for the exponential stability of a class of nonlinear differential equations are also presented. The obtained results generalize some existing results in the literature. Applications to neutral networks are investigated. Some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We propose new approximate alternating projection methods, based on randomized sketching, for the low-rank nonnegative matrix approximation problem: find a low-rank approximation of a nonnegative matrix that is nonnegative, but whose factors can be arbitrary. We calculate the computational complexities of the proposed methods and evaluate their performance in numerical experiments. The comparison with the known deterministic alternating projection methods shows that the randomized approaches are faster and exhibit similar convergence properties. [ABSTRACT FROM AUTHOR]

In this paper we present a class of convergent infinite products ∏ k = 1 ∞ H k = lim k → ∞ (H k H k − 1 ⋯ H 1) of 3 × 3 hollow matrices of the form H k := [ 0 a k a k b k 0 b k c k c k 0 ] , (0 < a k , b k , c k < 1 / 2). These sequences of hollow matrices are obtained from the Ando-Li-Mathias matrix geometric mean of 2 × 2 positive definite matrices. We further provide a three dimensional analytic manifold and an analytic self-map realizing such sequences. [ABSTRACT FROM AUTHOR]

We show how to construct nonnegative low-rank approximations of nonnegative tensors in Tucker and tensor train formats. We use alternating projections between the nonnegative orthant and the set of low-rank tensors, using STHOSVD and TTSVD algorithms, respectively, and further accelerate the alternating projections using randomized sketching. The numerical experiments on both synthetic data and hyperspectral images show the decay of the negative elements and that the error of the resulting approximation is close to the initial error obtained with STHOSVD and TTSVD. The proposed method for the Tucker case is superior to the previous ones in terms of computational complexity and decay of negative elements. The tensor train case, to the best of our knowledge, has not been studied before. [ABSTRACT FROM AUTHOR]

This survey contains a selection of topics unified by the concept of positivesemi-definiteness (of matrices or kernels), reflecting natural constraintsimposed on discrete data (graphs or networks) or continuous objects(probability or mass distributions). We put emphasis on entrywise operationswhich preserve positivity, in a variety of guises. Techniques from harmonicanalysis, function theory, operator theory, statistics, combinatorics, andgroup representations are invoked. Some partially forgotten classical roots inmetric geometry and distance transforms are presented with comments and fullbibliographical references. Modern applications to high-dimensional covarianceestimation and regularization are included.

Let ϕ be a positive map from the n × n matrices M n to the m × m matrices M m . It is known that ϕ is 2-positive if and only if for all K ∈ M n and all strictly positive X ∈ M n , ϕ (K ∗ X - 1 K) ⩾ ϕ (K) ∗ ϕ (X) - 1 ϕ (K) . This inequality is not generally true if ϕ is merely a Schwarz map. We show that the corresponding tracial inequality Tr [ ϕ (K ∗ X - 1 K) ] ⩾ Tr [ ϕ (K) ∗ ϕ (X) - 1 ϕ (K) ] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. [ABSTRACT FROM AUTHOR]

We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m ! ϕ m for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new version of a Schur–Horn type theorem. Meanwhile, we provide a new proof of a recent result in [Bhatia R, Jain T. A Schur-Horn theorem for symplectic eigenvalues. Linear Algebra Appl. 2020;599:133–139]. [ABSTRACT FROM AUTHOR]

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors. [ABSTRACT FROM AUTHOR]

The notions of C-tensor, C0-tensor and C ¯ -tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and C ¯ -tensor are provided. We next show that the sum of two C-tensors (C0-tensors) is a C-tensor (C0-tensor) while the Hadamard product of two C-tensors (C0-tensors) is not a C-tensor (C0-tensor). We also present a result that illustrates the Hadamard product of two C-tensor is again a C-tensor under some sufficient conditions. As an application of these classes of tensors, an exclusion interval for the real eigenvalues of a real tensor is proposed. Finally, we provide a necessary and sufficient condition for the exclusion interval to be nonempty. [ABSTRACT FROM AUTHOR]