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Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of unit-trace n × n PSD matrices, denoted by D, using at most N number of k × k PSD constraints?" In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any k = o (n) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have S + k -extension complexity at least exp (C · min { n , n / k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S + k -extension complexity at least exp (C · min { n 1 / 3 , n / k }) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any k = o (n / log 2 n) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of S + n , which corresponds to the special case with k = 1 . [ABSTRACT FROM AUTHOR]

A map that is defined on the set of all symmetric matrices over a field is rank-one nonincreasing if it maps the matrices of rank one to matrices of rank at most one. In the case of the fields with two or three elements nonstandard examples of such additive maps are known to exist. In this paper, all these maps are characterized. Moreover, the existence of nonstandard maps is understood in a geometric sense since they exist precisely in the two cases, where the whole n-dimensional vector space over can be written as a union of two hyperbolic/parabolic quadrics of low indices. This geometric interpretation enables us to realize that these weird examples represent just a tip of an iceberg that appears in an analogous problem on symmetric tensors, where nonstandard maps exist also over larger fields. A few open problems related to the multilinear analog and to homogeneous polynomials are posed. [ABSTRACT FROM AUTHOR]

Motivated by the work of Dahrouj and Yu in applying the Han-Kobayashi transmission strategy for mitigating the intercell interference in a multi-cell multi-user multiple-input single-output interference network (MISO IN), this paper considers splitting messages into private and common parts in a multi-cell multi-user MIMO IN. Specifically, the covariances of the private messages and common messages are designed to optimize either the sum rate or the minimal rate. The common messages and private messages are decoded in sequence using successive decoding. This paper shows how these difficult optimization problems can be adequately solved by means of d.c. (difference of concave functions) optimization over a simple convex set. Numerical and simulation results also reveal the great advantage of our proposed solutions for various types of INs. In particular, the proposed solutions are shown to outperform the algorithm developed by Dahrouj and Yu for the simpler case of the MISO IN. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a weighted alternating direction method of multipliers (ADMM) to solve the consensus optimization problem over a decentralized network. In the proposed algorithm, every node holds its local objective function, exchanges its current iterate with a subset of neighbors, carries on local computation, and eventually reaches an optimal and consensual solution that minimizes the summation of the local objective functions. Compared with the conventional ADMM that is popular in decentralized network optimization, the weighted ADMM is able to reduce the communication cost spent in the optimization process through tuning the weight matrices, which assign beliefs on the neighboring iterates. We first prove convergence and establish linear convergence rate of the weighted ADMM. Second, we maximize the derived convergence speed and obtain the best weight matrices on a given topology. Third, observing that exchanging information with all the neighbors is expensive, we maximize the convergence speed while limit the number of communication arcs. This strategy finds a subset of arcs within the underlying topology to fulfill the optimization task while leads to a favorable tradeoff between the number of iterations and the communication cost per iteration. Numerical experiments demonstrate advantages of the weighted ADMM over its conventional counterpart in expediting the convergence speed and reducing the communication cost. [ABSTRACT FROM PUBLISHER]

In a previous paper, we completely described cyclotomic matrices—integer symmetric matrices of spectral radius at most 2. In this paper, we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular, we solve the strong version of Lehmer’s problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least ‘Lehmer’s number’ λ0=1.17628…. [ABSTRACT FROM PUBLISHER]

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: one is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data. [Copyright &y& Elsevier]

Recently several authors investigated structures of generalized bi-circular projections in spaces where the descriptions of the group of surjective isometries are known. Following the same idea in this paper we give complete descriptions of generalized 3-circular projections for symmetric norms on ℂn and Mm×n(ℂ). [ABSTRACT FROM AUTHOR]

Abstract: The fourth moment of a random vector is a matrix whose elements are all moments of order four which can be obtained from the random vector itself. In this paper, we give a lower bound for its dominant eigenvalue and show that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined in more detail. [Copyright &y& Elsevier]

In their article in the journal Mathematical Reflections, Campos Daniel and Verdiyan Verdan (2009) proposed an interesting and difficult inequality, which has appeared on several occasions in the MathLinks forum (http://www.artofproblemsolving.com/Forum/index.php), without any references to its solution. In this paper, we present two original solutions to this inequality problem. We also consider various generalizations of this inequality from various perspectives. Finally, we apply the general results to solve some other difficult inequalities. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct a modified Green potential in the upper-half space of the n-dimensional Euclidean space. Meanwhile, the behavior at infinity of it is also given. [Copyright &y& Elsevier]

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying $${Du \in K}$$ . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each $${F \in {\rm int}\,K^{rc}}$$ there exists a Lipschitz mapping u satisfying Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal. [ABSTRACT FROM AUTHOR]

Abstract: A well-known property of an -matrix is that the inverse is element-wise non-negative, which we write as . In this paper, we consider element-wise perturbations of non-symmetric tridiagonal -matrices and obtain bounds on the perturbations so that the non-negative inverse persists. Sufficient bounds are written in terms of decay estimates which characterize the decay of the elements of the inverse of the unperturbed matrix. Results for general symmetric matrices and symmetric Toeplitz matrices are obtained as special cases and compared with known results. [Copyright &y& Elsevier]

Abstract: An matrix A is said to be -symmetric if for given . In this paper, the following -symmetric Procrustes problem is studied. Find the -symmetric matrix A which minimizes the Frobenius norm of , where X and B are given rectangular matrices. We use Project Theorem, the singular-value decomposition and the generalized singular-value decomposition of matrices to analysis the problem and to derive a stable method for its solution. The related optimal approximation problem to a given matrix on the solution set is solved. Furthermore, the algorithm to compute the optimal approximate solution and the numerical experiment are given. [Copyright &y& Elsevier]

For α ϵ]0, π/2[, let (Σ)α be the control system ẋ = (F +uG)x, where x belongs to the two-dimensional unit sphere S2, u ϵ [-1, 1], and F,G are 3×3 skew-symmetric matrices generating rotations with perpendicular axes and of respective norms cos(α) and sin(α). In this paper, we study the time optimal synthesis (TOS) from the north pole (0, 0, 1)T associated to (ϵ)α, as the parameter a tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a "two-snakes" configuration on the whole S2, except for a neighborhood Uα of the south pole (0, 0,-1)T of diameter at most O(α). We next show that, inside Uα, the TOS depends on the relationship between r(α) := π/2α - [π/2α] and a. More precisely, we characterize three main relationships by considering sequences (αk)k=0 satisfying (α) r(αk) = r, (b) r(αk) = Cαk, and (c) r(αk) = 0, where r ϵ (0, 1) and C > 0. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as a tends to zero, of the corresponding TOS inside Uα. [ABSTRACT FROM AUTHOR]

This paper presents a divide-and-conquer method for computing the symmetric singular value decomposition, or Takagi factorization, of a complex symmetric and tridiagonal matrix. An analysis of accuracy shows that our method produces accurate Takagi values and orthogonal Takagi vectors. Our preliminary numerical experiments have confirmed our analysis and demonstrated that our divide-and-conquer method is much more efficient than the implicit QR method even for moderately large matrices. [ABSTRACT FROM AUTHOR]

Abstract: Let be a nontrivial involution, i.e., and . A matrix is called R-symmetric if . The solvability conditions and the expression of the matrix inverse problem for R-symmetric matrices with are derived, also the least-squares solutions of the matrix inverse problem for R-symmetric matrices with are given. The corresponding optimal approximation problem for R-symmetric matrices with is considered. We firstly point out that the optimal approximation problem is solvable, then get the expression of its unique solution. It can be seen that this paper generalizes the results mentioned in Zhou [F.-Z. Zhou, L. Zhang, X.-Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581–1589]. [Copyright &y& Elsevier]

Abstract: We study in this paper several properties of the eigenvalues function of a Euclidean Jordan algebra, extending several known results in the framework of symmetric matrices. In particular, we give a concise form for the directional differential of a single eigenvalue. We especially focus on spectral functions F on Euclidean Jordan algebras, which are the composition of a symmetric real-valued function f with the eigenvalues function. We explore several properties of f that are transferred to F, in particular convexity, strong convexity and differentiability. Spectral mappings are also considered, a special case of which is the gradient mapping of a spectral function. Answering a problem proposed by H. Sendov, we give a formula for the Jacobian of these functions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209–226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues. [Copyright &y& Elsevier]

Abstract: The set of real matrices described by a sign pattern (a matrix whose entries are elements of {+,−,0}) has been studied extensively but only loose bounds were available for the minimum rank of a tree sign pattern. A simple graph has been associated with the set of symmetric matrices having a zero–nonzero pattern of off-diagonal entries described by the graph, and the minimum rank/maximum eigenvalue multiplicity among matrices in this set is readily computable for a tree. In this paper, we extend techniques for trees to tree sign patterns and trees allowing loops (with the presence or absence of loops describing the zero–nonzero pattern of the diagonal), allowing precise computation of the minimum rank of a tree sign pattern and a tree allowing loops. For a symmetric tree sign pattern or a tree that allows loops, we provide an algorithm that allows exact computation of maximum multiplicity and minimum rank, and can be used to obtain a symmetric integer matrix realizing minimum rank. [Copyright &y& Elsevier]

In this paper, we introduce a fast algorithm which can be used to determine the triangular factorization of an inverse of a symmetric matrix of Loewner type by using O(n²) operations. [ABSTRACT FROM AUTHOR]

Suppose F is a field different from F2, the field with two elements. Let Mn (F) and Sn (F) be the space of n × n full matrices and the space of n × n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn (F), Mn (F)} , we say that a linear map f from G1 to G2 is inverse-preserving if f(X)-1 = f(X-1) for every invertible X ∈ G1. Let ... (G1, G2) denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets ... (Sn (F), Mn (F)), ... (Sn (F), Sn (F)), ... (Mn (F), Mn(F)) and are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the weak solutions $$ u \in W^{{1,p}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {1 < p < \infty } \right)} $$ of the following equation with vanishing mean oscillation coefficients A( x): belong to $$ W^{{1,q}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {\forall q \in {\left( {p,\infty } \right)}} \right)} $$, provided $$ F \in L^{q}_{{{\text{loc}}}} {\left( \Omega \right)} $$ and B( x, u, h) satisfies proper growth conditions, where Ω ⊂ R N ( N ≥ 2) is a bounded open set, A( x) = ( A ij( x)) N× N is a symmetric matrix function. [ABSTRACT FROM AUTHOR]

Let A and K be real symmetric matrices with K2 = I. In the article ’A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices’ [D. Tao and M. Yasuda, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 885–895], it was shown that (1) AK = KA if and only if the spectrum of A equals the spectrum of KA up to sign and (2) AK = -KA if and only if the spectrum of A equals the spectrum of KA multiplied by i. This paper extends these spectral characterizations from the case of real symmetric matrices to that of self-adjoint compact linear operators in a complex Hilbert space. Some consequences of these results are mentioned, including an application that describes the correspondence between the spectrum of a real symmetric Toeplitz matrix T and its associated Hankel matrix JT, where J is the so-called exchange matrix. [ABSTRACT FROM AUTHOR]

In the convergence theory of multisplittings for symmetric positive definite (s.p.d.) matrices it is usually assumed that the weighting matrices are scalar matrices, i.e., multiples of the identity. In this paper, this restrictive condition is eliminated. In its place it is assumed that more than one (inner) iteration is performed in each processor (or block). The theory developed here is applied to nonstationary multisplittings for s.p.d. matrices, as well as to two-stage multisplittings for symmetric positive semidefinite matrices. [ABSTRACT FROM AUTHOR]

In this paper, we propose a restarted variant of the Lanczos method for symmetric eigenvalue problems named the thick-restart Lanczos method. This new variant is able to retain an arbitrary number of Ritz vectors from the previous iterations with a minimal restarting cost. Since it restarts with Ritz vectors, it is simpler than similar methods, such as the implicitly restarted Lanczos method. We carefully examine the effects of the floating-point round-off errors on stability of the new algorithm and present an implementation of the partial reorthogonalization scheme that guarantees accurate Ritz values with a minimal amount of reorthogonalization. We also show a number of heuristics on deciding which Ritz pairs to save during restart in order to maximize the overall performance of the thick-restart Lanczos method. [ABSTRACT FROM AUTHOR]

• In the abstract, the correct definition of A ex (G) is added. • The definitions for the matrix A and for the matrix SDD are added. • The adjacency matrix A (G) was defined. • On page 2, after "For recent research along these lines, we added: see [5,20,33] and "Recall that the extended adjacency matrix is just one among a large number of degree-based graph matrices; for details see [A]." • On page 10, together with Refs. [17, 21, 19] we quoted also [B]:r • On page 10, together with Ref. [27] we quoted also Ref [A]. • Two following refs are added: [A] K. C. Das, I. Gutman, I. Milovanovi'c, E. Milovanovi'c, B. Furtula, Degree-based energies of graphs, Linear Algebra Appl. 554 (2018) 185–204. [B] I. Gutman, H. Ramane, Research on graph energies in 2019, MATCH Commun. Math. Comput. Chem. 84 (2020) 277–292. The extended adjacency matrix of graph G , A e x is a symmetric real matrix that if i ≠ j and u i u j ∈ E (G) , then the i j th entry is d u i 2 + d u j 2 / 2 d u i d u j , and zero otherwise, where d u indicates the degree of vertex u. In the present paper, several investigations of the extended adjacency matrix are undertaken and then some spectral properties of A e x are given. Moreover, we present some lower and upper bounds on extended adjacency spectral radii of graphs. Besides, we also study the behavior of the extended adjacency energy of a graph G. [ABSTRACT FROM AUTHOR]

In this short paper, practical estimates for the extreme eigenvalues of symmetric matrices and their inverses are obtained, and a criterion for Stieltjes matrices is presented. [ABSTRACT FROM AUTHOR]