Generalized marginal problem in terms of multi-system Hermitian and positive semidefinite matrices

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

Maximum permanent and Hermitian matrices

Norms and inequalities for condition numbers, III

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

Sparse matrix decompositions and graph characterizations

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

An inertia theorem for Lyapunov's equation and the dimension of a controllability space

In this article, we obtain another proof of the following classical trace inequality which says that if A is a positive semidefinite matrix and B is a Hermitian matrix, then tr A? BA?B ? tr A? BA?B for all non-negative real numbers ?, ?, ?, ? for which ? + ? = ? + ? and max {?,?} ? max {?,?}.This is a generalization of trace inequalities due to T. Ando, F.Hiai, and K. Okubo for the special cases when ?= ?+?,?=0 and when ?= ?=?+?/2, namely tr (A?+?/2B)2 ? tr A?BA?B? tr A?+?B2.

In this study, we consider a non-linear matrix equation of the formwhere Q is a Hermitian positive definite matrix, A * i stands for the conjugate transpose of an n n matrix A i and F j 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.

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

It is proven that the matrices AB and BA formed from the product a positive definite self-adjoint matrix A and a self-adjoint matrix B has real eigenvalues and a complete set of eigenvectors. If B is positive (negative) semidefinite the eigenvalues are greater (less) than or equal to zero. These properties have been useful in the analysis of multicomponent diffusion and distillation processes.

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let $$\varepsilon$$ and $$\varepsilon_{+}$$ be a finite dimensional real vector space and a symmetric cone embedded in $$\varepsilon$$; examples of $$\varepsilon$$ and $$\varepsilon_{+}$$ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over $$\varepsilon_{+}$$, i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region $$\{ {\bf x} : b({\bf x}) \in \varepsilon_{+}\}$$, where b(x) denotes an $$\varepsilon$$- valued polynomial in x. It is shown under a certain moderate assumption on the $$\varepsilon$$-valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem.