Matrix Trace Inequalities Research Articles

Impulse Controllability for Singular Hybrid Coupled Systems

This study examines the concept of impulse controllability within singular hybrid coupled systems through the utilisation of decentralised proportional plus derivative (P-D) output feedback. By employing the Differential Mean Value Theorem, the nonlinear model can be converted into a linear parameter-varying large-scale system. Our analysis leads to the establishment of algebraic conditions that are both necessary and sufficient for the existence of a decentralised P-D output feedback controller that can guarantee impulse controllability in these complex systems. Moreover, we address the issue of admissibility within these systems by employing matrix trace inequalities. We present a novel sufficient condition for impulse controllability, which offers a new perspective on addressing this challenging problem. To validate our findings, we present numerical examples that demonstrate the effectiveness of the proposed methodologies in practice.

On a Matrix Trace Inequality for Hermitian Matrices

On a Matrix Trace Inequality for Hermitian Matrices

Matrix trace inequalities related to the Tsallis relative entropies of real order, II

In this paper, as a continuation of two previous papers, we show matrix trace inequalities related to two Tsallis relative entropies of all real order: For density matrices ρ and σ, and each α∈R﹨{0}, the Tsallis relative entropy Dα(ρ|σ) is defined by Dα(ρ|σ)=−Tr(ρ1−ασα−ρα) and NTα(ρ|σ) is defined by NTα(ρ|σ)=−Tr[ρ♮ασ−ρα], where ♮α is defined by ρ♮ασ=ρ1/2(ρ−1/2σρ−1/2)αρ1/2. Then we show the order relation between two Tsallis relative entropies Dα and NTα of all real order α∈R and a 1-parameter extension of the path connecting them of all real order. Moreover, we show estimates of the difference between two Tsallis relative entropies of all real order.

Entanglement Monogamy via Multivariate Trace Inequalities

Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.

Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.

Jointly convex mappings related to Lieb’s theorem and Minkowski type operator inequalities

Employing the notion of operator log-convexity, we study joint concavity/convexity of multivariable operator functions related to Lieb’s theorem: $$(A,B)\mapsto F(A,B)=\psi \left[ \Phi (f(A))\ \sigma \ \Psi (g(B))\right] $$ , where $$\Phi $$ and $$\Psi $$ are positive linear mappings and $$\sigma $$ is an operator mean. As applications, we prove jointly concavity/convexity of matrix trace functions $$\mathrm {Tr}\left\{ F(A,B)\right\} $$ . Moreover, considering positive multi-linear mappings in F(A, B), our study of the joint concavity/convexity of $$(A_1,\ldots ,A_k)\mapsto \psi \left[ \Phi (f(A_1),\ldots ,f(A_k))\right] $$ provides some generalizations and complement to results of Ando and Lieb concerning the concavity/convexity of mappings involving tensor product. In addition, we present Minkowski type operator inequalities for a unital positive linear mapping, which is an operator version of Minkowski type matrix trace inequalities under a more general setting than Carlen and Lieb, Bekjan, and Hiai.

Matrix trace inequalities related to the Tsallis relative entropies of real order

In this paper, we show matrix trace inequalities related to the Tsallis relative entropy of real order: For positive definite matrices ρ and σ, and each 0<α≤1Dα(ρ|σ)≤−Tr[ρ1−qqTαq(ρq|σq)] for all q≥α>0, where the Tsallis relative entropy Dα(ρ|σ) is defined by Dα(ρ|σ)=−Tr(ρ1−ασα−ρα) and the Tsallis relative operator entropy Tα(ρ|σ) is defined by Tα(ρ|σ)=ρ♯ασ−ρα, where ♯α is the matrix α-geometric mean. Moreover, we show estimates of the difference between two Tsallis relative entropies of real order.

Quantum $$\alpha $$-fidelity of unitary orbits

The extremum values of quantum $$\alpha $$ -fidelity under unitary orbits of quantum states are explicitly derived by applying rearrangement inequalities, matrix trace inequalities, and theory of majorization. Furthermore, the $$\alpha $$ -fidelity is successfully verified to go through the whole closed interval, which works from the minimum value to the maximum value.

Matrix Poincaré inequalities and concentration

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

A matrix trace inequality

A matrix trace inequality

Matrix trace inequalities on Tsallis relative entropy of negative order

In this paper, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive definite matrices. Moreover, we show an estimate of the difference between two Tsallis relative entropies of negative order.

An Introduction to Kristof’s Theorem for Solving Least-Square Optimization Problems Without Calculus

ABSTRACTKristof’s Theorem (Kristof, 1970) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof’s Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof’s (1964, 1970) writings. In this article, I describe the underlying logic of Kristof’s Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem’s proof. I then show how Kristof’s Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017) code to perform the calculations described in the text.

English

English

On Tricomplex Representation Methods of A Matrix Trace Inequality

On Tricomplex Representation Methods of A Matrix Trace Inequality

On the Araki–Lieb–Thirring inequality in the semifinite von Neumann algebra

This paper extends a recent matrix trace inequality of Bourin–Lee to semifinite von Neumann algebras. This provides a generalization of the Lieb–Thirring-type inequality in von Neumann algebras due to Kosaki. Some new inequalities, even in the matrix case, are also given for the Heinz means.

On a matrix trace inequality due to Ando, Hiai and Okubo

Ando et al. have proved that inequality \(\Re \mathfrak{e}trA^{p_1 } B^{q_1 \ldots } A^{p_k } B^{q_k } \leqslant trA^{p_1 + \ldots + p_k } B^{q_1 + \ldots + q_k }p\) is valid for all positive semidefinite matrices A,B and those nonnegative real numbers p1, q1,..., pk, qk which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of p1, q1,..., pk, qk ≥ 0. We also study related trace inequalities.

A note on the inverse spectral problem for symmetric doubly stochastic matrices

The symmetric doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real -tuple to be the spectrum of an symmetric doubly stochastic matrix. For , this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for this problem using some matrix trace inequalities. In addition, we prove that this family of new inequalities sharpen the existing known necessary conditions for the inverse spectral problem of nonnegative matrices. Finally, we prove that these necessary conditions are not sufficient for the case .

Several matrix trace inequalities on Hermitian and skew-Hermitian matrices

In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO).

Matrix and Operator Trace Inequalities

Matrix and Operator Trace Inequalities

Tsallis entropies and matrix trace inequalities in quantum statistical mechanics

Extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the Tsallis entropy. Namely, one-parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized Peierls–Bogoliubov inequality is stated, in the sense that one may be obtained from the other, and vice versa.