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This note offers some remarks on a norm version of the matrix arithmetic–geometric inequality. [ABSTRACT FROM AUTHOR]

In this paper, we give a matrix version of an equivalent form of the classical arithmetic–geometric mean inequality for two positive scalars. Applications and generalizations of our results are also given. [ABSTRACT FROM AUTHOR]

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we provided an elementary proof of a multiple Hlawka type inequality proved by W. Berndt and S. Sra in [1]. Our proof does not use the tensor product method. [ABSTRACT FROM AUTHOR]

In this short note, we presented a number of alternative explicit necessary and sufficient conditions for diagonal stability along with a new proof of a well-known result in this regard. [ABSTRACT FROM AUTHOR]

In this short note, we presented a number of alternative explicit necessary and sufficient conditions for diagonal stability along with a new proof of a well-known result in this regard.

Using the Khatri-Rao product, we presented new characterizations for the common Lyapunov diagonal stability for a family of real matrices $ \mathcal{A} $. For special partitions $ \alpha $, we used the notion of $ \mathcal{P}^{\alpha} $-sets and common $ \alpha $-scalar Lyapunov stability to formulate further characterizations. Furthermore, generalizations of these results to the common $ \alpha $-scalar Lyapunov stability were developed. Our goal of this paper was to unify and enhance relevant work.

Using the Khatri-Rao product, we presented new characterizations for the common Lyapunov diagonal stability for a family of real matrices A. For special partitions a, we used the notion of Pα-sets and common α-scalar Lyapunov stability to formulate further characterizations. Furthermore, generalizations of these results to the common α-scalar Lyapunov stability were developed. Our goal of this paper was to unify and enhance relevant work. [ABSTRACT FROM AUTHOR]

We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) $ \sharp $ and semi-tensor products $ \ltimes $. Indeed, for each real number $ t $ and two positive definite matrices $ A $ and $ B $ of arbitrary sizes, the $ t $-weighted SGM $ A \, \diamondsuit_t \, B $ of $ A $ and $ B $ is a unique positive solution $ X $ of the equation $ A^{-1}\,\sharp\, X \; = \; (A^{-1}\,\sharp\, B)^t. $ We then established fundamental properties of the weighted SGMs based on MGMs. In addition, $ (A \, \diamondsuit_{1/2} \, B)^2 $ is positively similar to $ A \ltimes B $ and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.

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]

Let G be a connected graph on n vertices with adjacency matrix A G. Associated to G is a polynomial d G (x 1 , ... , x n) of degree n in n variables, obtained as the determinant of the matrix M G (x 1 , ... , x n) , where M G = Diag (x 1 , ... , x n) − A G. We investigate in this article the set V d G (r) of non-negative values taken by this polynomial when x 1 , ... , x n ≥ r ≥ 1. We show that V d G (1) = Z ≥ 0. We show that for a large class of graphs one also has V d G (2) = Z ≥ 0. When V d G (2) ≠ Z ≥ 0 , we show that for many graphs V d G (2) is dense in Z ≥ 0. We give numerical evidence that in many cases, the complement of V d G (2) in Z ≥ 0 might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial. [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]

We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the parametrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is linearity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite matrices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry. We show that their linearity problems for linearly independent A and B are equivalent to the solvability of the following equations for positive reals x and y with x y < 1 4 respectively (x A + y B) − 1 = y A − 1 + x B − 1 , (A B) 1 / 2 + (B A) 1 / 2 2 = x A + y B. [ABSTRACT FROM AUTHOR]

The Cayley transform of a matrix A , C (A) = (I + A) − 1 (I − A) , has been investigated in the contexts of skew-symmetry, eigenvalue mapping and as a factorization technique for several matrix positivity classes. Here, we extend the discussion to Cayley transforms of almost skew-Hermitian matrices, positive definite matrices, semipositive matrices, as well as subclasses of the M -matrices, H -matrices and their inverses. [ABSTRACT FROM AUTHOR]

A conjecture of Dinh, Ahsani, and Tam, was recently settled in [7]. In this note, we give a refinement to that result, namely if Ai and Bi are positive definite matrices and Z = [Zij] is the block matrix such that Zij = Bi 1/2 (m/Σ/k=1 Ak) B1/2 j for all i,j = 1, ...,m, then |||m/Σ/i=1 ( A²i # B² i )r ||| ≤ ||| Zr ||| ≤ ||| ((m/Σ/i=1 Ai)rp/2 (m/Σ/i=1 B²)rp (m/Σ/i=1 Ai)rp/2 )1/p |||, for all unitarily invariant norms, for all p > 0 and r ≥ 1 such that rp ≥ 1. Our approach provides us with an alternative proof without using the method of majorization that was used in [7]. As a byproduct, we get a refinement to a result of Audenaert in 2015. [ABSTRACT FROM AUTHOR]

Let r be any real number and for any n let p1,...., pn be distinct positive numbers. A Kwong matrix is the n x n matrix whose (i, j) entry is (pri +prj)/(pi +pj). We determine the signatures of eigenvalues of all such matrices(The corresponding problem for the family of Loewner matrices [(pri +prj)/(pi - pj)] has been solved earlier. [ABSTRACT FROM AUTHOR]

We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.

We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters. [ABSTRACT FROM AUTHOR]

For solving horizontal linear complementarity problem (HLCP), we propose a nonlinear lopsided PSS-like modulus-based matrix splitting iteration method and a new nonlinear lopsided PSS-like modulus-based matrix splitting iteration method according to the characteristics of the HLCP. The proposed methods contain the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method and a new nonlinear lopsided HSS-like modulus-based matrix splitting iteration method. When the system matrices are positive definite matrices, we analyze the proposed methods in theory and prove the convergence of these methods. Numerical experiments in this paper demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We investigate the Riccati matrix equation XA-1X = B in which the conventional matrix products are generalized to the semi-tensor products n. When A and B are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of A and B. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices. [ABSTRACT FROM AUTHOR]

Let A and B be positive definite matrices of the same size, and let α ≥ 1. If the spectrum of A B − 1 lies in the interval [ 1 , α ] , then | | | A t B 1 − t + B t A 1 − t | | | ≤ α | | | A + B | | | for t ∈ [ 0 , 1 ] and for every unitarily invariant norm. This norm inequality is closely related to an open question of Bourin. [ABSTRACT FROM AUTHOR]

In this paper, we show how the outer bounds of the extremal eigenvalues of real positive definite symmetric irreducible matrices may be improved by the simple application of optimizing the Gerschgorin bounds. The method is easy to apply and yields fairly accurate results with minimal effort. [ABSTRACT FROM AUTHOR]

One of the most well-known methods for unconstrained problems is the quasi-Newton approach, iterative solutions. The great precision and quick convergence of the quasi-Newton methods are well recognized. In this work, the new algorithm for the symmetric rank one SR1 method is driven. The strong Wolfe line search criteria define the step length selection. We also proved the new quasi-Newton equation and positive definite matrix theorem. Preliminary computer testing on the set of fourteen unrestricted optimization test functions leads to the conclusion that this new method is more effective and durable than the implementation of classical SR1 method in terms of iterations count and functions. [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 study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (also conjectured by Erdős and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph G$$ G $$ homogeneous if the marginal distributions on the edges are all identical. We show that if G$$ G $$ is bipartite, then the differential entropy of any homogeneous GMRF on G$$ G $$ is at least |E(G)|$$ \mid E(G)\mid $$ times the edge entropy plus |V(G)|−2|E(G)|$$ \mid V(G)\mid -2\mid E(G)\mid $$ times the point entropy. We also show that in the case of non‐negative correlation on edges, the result holds for an arbitrary graph G$$ G $$. The connection between Sidorenko's conjecture and GMRF's is established via a large deviation principle on high dimensional spheres combined with graph limit theory. It is also observed that the system we study exhibits a phase transition on large girth regular graphs. Connection with Ihara zeta function and the number of spanning trees is also discussed. [ABSTRACT FROM AUTHOR]

Abstract In this study, we consider a nonlinear matrix equation of the form X = Q + ∑ i = 1 m A i ∗ G ( X ) A i $\mathcal{X}= \mathcal{Q} + \sum_{i=1}^{m} \mathcal{A}_{i}^{*} \mathcal{G} (\mathcal{X})\mathcal{A}_{i}$ , where Q $\mathcal{Q}$ is a Hermitian positive definite matrix, A i ∗ $\mathcal{A}_{i}^{*}$ stands for the conjugate transpose of an n × n $n\times n$ matrix A i $\mathcal{A}_{i}$ , and G $\mathcal{G}$ is an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that G ( O ) = O $\mathcal{G}(O)=O$ . We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation. For this, we derive some fixed point results for Suzuki-FG contractive mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order). We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of nonlinear matrix equations through an illustration with graphical representations.

In this paper, we first give an alternative proof for a result of Liu et al. in [Math. Inequal. Appl. 20 (2017) 537–542]. Then we present two inequalities for the block Hadamard product and the Khatri-Rao product respectively. The inequalities obtained extend the result of Liu et al.

One of the most well-known methods for unconstrained problems is the quasi-Newton approach, iterative solutions. The great precision and quick convergence of the quasi-Newton methods are well recognized. In this work, the new algorithm for the symmetric rank one SR1 method is driven. The strong Wolfe line search criteria define the step length selection. We also proved the new quasi-Newton equation and positive definite matrix theorem. Preliminary computer testing on the set of fourteen unrestricted optimization test functions leads to the conclusion that this new method is more effective and durable than the implementation of classical SR1 method in terms of iterations count and functions.

In this study, we consider a non-linear matrix equation of the form X = Q + Σi=1m Σj=1t Ai Fj (X) Ai where Q is a Hermitian positive definite matrix, Ai* stands for the conjugate transpose of an n × n matrix Ai and Fj are order-preserving continuous mappings from the set of all Hermitian matrices to the set of all positive definite matrices such that F(O)=O. We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation. For this, we derive some fixed point results for Suzuki-implicit type mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order). We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of non-linear matrix equations. [ABSTRACT FROM AUTHOR]

In this paper we present a class of Riemannian equilateral triangles Δ A B C of the Cartan-Hadamard Riemannian manifold of N × N positive definite matrices of determinant 1 whose Karcher mean of vertices is of the form Λ (A , B , C) = A + B + C det ⁡ (A + B + C) N . The obtained class of triples (A , B , C) forms an analytic manifold with an analytic parameterization over SL (N , C) ± × R. The determinantal identity det ⁡ (A + B + C) = det ⁡ (A − 1 + B − 1 + C − 1) and an extended arithmetic-Karcher-harmonic mean inequalities is derived for such triples. [ABSTRACT FROM AUTHOR]

Consider the following problem: given a positive semidefinite (definite) matrix A ∈ Cmxm, and B ∈ Cnxm, such that ... = μ, find a matrix X satisfying subject to .... In 1996, Zheng presented the expressions for X formed by Löwner partial ordering when the problem is solvable, and pointed out that there may be no X satisfying these expressions. In this paper, we present some sufficient and necessary conditions for the existence of X and provide its explicit general expressions. [ABSTRACT FROM AUTHOR]

For the large sparse linear complementarity problem, by reformulating it as implicit fixed-point equations, we establish a modulus-based iteration method with the aid of an inexact non-alternating preconditioned matrix splitting iteration method used as an inner iteration to solve the module equations rapidly in each iteration step. The convergence properties of this method are carefully demonstrated under certain conditions. Numerical results, including the numbers of iteration steps and the computing times, validate that our method is superior to the other three iteration methods. [ABSTRACT FROM AUTHOR]

Let Ai, P, i = 1, ..., m, be n by n complex matrices such that each Ai is positive definite. It is shown, among other inequalities, that If 0 < aj ≤ sn(Aj), C m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) a j and P is Hermitian, then for every unitarily invariant norm ∥ |. ∥ | , ∥ | C m P + P C m ∥ | ≥ α ∥ | P ∥ | , where α = max 2 m − 1 , m 1 + min a 1 2 , ... , a m 2 . If s1(Aj) ≤ bj, ∑ i = 1 m b i = 1 and D m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) b j , then for every unitarily invariant norm ∥ |. ∥ | , 2 m P ≥ ∥ | D m P + P D m ∥ |. [ABSTRACT FROM AUTHOR]

The Popoviciu inequality is known as a weighted version of the celebrated Aczel inequality. In this paper, we establish an analogue of the Popoviciu inequality for the determinants of positive definite matrices, and then we extend our result to the case where the underlying matrices are sectorial matrices, i.e. matrices whose field of values (or numerical range) are contained in a sector. [ABSTRACT FROM AUTHOR]

We consider the Lie semigroup of symplectic Hamiltonians acting on the open convex cone of positive definite matrices via linear fractional transformations. Each member of its interior contracts strictly the invariant Finsler metric, the Thompson metric on the cone, and has a unique positive definite fixed point. We show that the fixed point map is smooth. As applications, we obtain the smooth dependence of the solutions of discrete algebraic Riccati equations and a family of smooth maps from the Siegel upper half-plane over the cone of positive definite matrices into its imaginary part. [ABSTRACT FROM AUTHOR]

In this paper we give some new power-type refinements and reverses of the weighted arithmetic–geometric mean inequality. Our result is a remarkable generalization of the one due to Furuichi (J Math Inequal 5(1):21–31, 2011), Manasrah and Kittaneh (J Math Anal Appl 361(1):262–269, 2010, Linear Multilinear Algebra 59(9):1031–1037, 2011). When the power is a positive integer number, our result is also a refinement of a very recent generalization established by Ighachane et al. (Math Inequal Appl 23(3):1079–1085, 2020) for this inequality. As an application, we provide some generalized inequalities for determinants of positive definite matrices. [ABSTRACT FROM AUTHOR]

The goal of this study is to solve a non-linear matrix equation of the form $ \mathcal{X} = \mathcal{Q} + \sum\limits_{i = 1}^{m} \mathcal{B}_{i}^{*}\mathcal{G} (\mathcal{X})\mathcal{B}_{i} $, where $ \mathcal{Q} $ is a Hermitian positive definite matrix, $ \mathcal{B}_{i}^{*} $ stands for the conjugate transpose of an $ n\times n $ matrix $ \mathcal{B}_{i} $ and $ \mathcal{G} $ an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that $ \mathcal{G}(O) = O $. We explore the necessary and sufficient criteria for the existence of a unique positive definite solution to a particular matrix problem. For the said reason, we develop some fixed point results for $ \mathcal{FG} $-contractive mappings on complete metric spaces equipped with any binary relation (not necessarily a partial order). We give adequate examples to confirm the fixed-point results and compare them to early studies, as well as four instances that show the convergence analysis of non-linear matrix equations using graphical representations.

In this work the positive semi-definiteness of symmetric matrices with non-negative entries and positive entries on the diagonal will be analyzed by using the Yager's family of t-norms. It can always be assumed that such an n × n matrix corresponds to a reflexive and symmetric fuzzy relation A on a set of cardinality n and the main result of this paper states that if A is transitive with respect to a specific t-norm T λ of Yager's family with λ depending on n , then A is positive semi-definite. This constitutes a surprising novel application of t-norms to an a priori unrelated field such as linear algebra. The result will be applied to give alternative proofs of the following two important facts. • Every min-indistinguishability operator on a finite set has a positive semi-definite matrix. • Every pseudoultrametric on a finite set is Euclidean. The reciprocal of the main result is not true but some results concerning this issue are studied. [ABSTRACT FROM AUTHOR]

By making use of the weighted geometric mean of positive definite matrices, we extend Hardy-Hilbert type inequalities to matrix case. Our results complement those of Hansen [Internat. J. Math. 21 (2010), no. 10, 1283-1295] and Kian [Ann. Funct. Anal. 3 (2012), no. 2, 128-134]. [ABSTRACT FROM AUTHOR]

We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions. [ABSTRACT FROM AUTHOR]

The article discusses extensions of the results related to bounds for the eigenvalues and spreads of matrices using traces.

In this paper, by transforming the quasi-complementarity problem (QCP) into an equivalent fixed-point format, we develop an efficient modulus-based matrix splitting (EMMS) iteration method for the large sparse QCPs. Compared with the MMS (Appl. Numer. Math. 55:8, 2018) and the TMMS (Comput. Appl. Math. 39:11, 2020) methods, the EMMS iteration method does not involve inner iterations and its initial vector can be chosen unlimitedly. Thus, the EMMS method may be more efficient than some existing ones in the practical calculation. And we prove that the EMMS iteration method is convergent under mild conditions when the system matrices are either positive definite matrices or H + -matrices, which generalize some known corresponding results. Besides, the convergence of the EM accelerated overrelaxation (EMAOR) method has also been analyzed, which is practical in real calculation. Lastly, several numerical examples are provided to show that the EMMS method is effective and feasible. [ABSTRACT FROM AUTHOR]