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We prove some inequalities involving fourth central moment of a random variable that takes values in a given finite interval. Both discrete and continuous cases are considered. Bounds for the spread are obtained when a given $$n\times n$$ complex matrix has real eigenvalues. Likewise, we discuss bounds for the spans of polynomial equations. [ABSTRACT FROM AUTHOR]

We prove a generalization of Samuelson's inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex n × n matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations. [ABSTRACT FROM AUTHOR]

Without 'positive definiteness' demanded in the present papers, the forward and reverse inequalities for Hadamard products of any number of invertible Hermitian matrices are obtained, and the sufficient and necessary conditions for the equations in these inequalities are given. As Hermitian positive matrices naturally satisfy the added constraints, these results generalize and improve the corresponding results in the present papers. Beyond that, with no demand of 'positive definiteness', these forward and backward inequalities are not determined mutually any longer. [ABSTRACT FROM AUTHOR]

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the 'qubit portrait' bound for the distance between the qutrit states are considered in explicit form. [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

Abstract: We show some majorization inequalities and apply them to derive norm, eigenvalue, singular value, and trace inequalities of matrices. We also present a generalization and a different proof of a recent result on majorization concerning partitioned Hermitian positive semidefinite matrices. [Copyright &y& Elsevier]

Abstract: This paper gives a group of expansion formulas for the inertias of Hermitian matrix polynomials , and through some congruence transformations for block matrices, where A is a Hermitian matrix. Then, the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and Hermitian unitary matrices. As applications, the paper establishes necessary and sufficient conditions for many matrix equalities to hold, as well as many inequalities in the Löwner partial ordering to hold. [Copyright &y& Elsevier]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain some new inequalities by means of the mean inequalities of random variables, which include generalizations of the Greub–Rheinboldt inequality. [Copyright &y& Elsevier]

Abstract: In this paper, an extended projection method in the complex number field is presented. We consider two classes of complex linear matrix inequalities and then derive the corresponding projection operators. Applications to the control system with pole assignment problem and the robust stability of linear descriptor systems, which are described in complex linear matrix inequalities, are given. Based on the numerical algorithms, some examples are illustrated for the merits of the proposed method. [Copyright &y& Elsevier]

The article discusses the proof of Thompson's determinantal inequality and tackles the conjugate transpose of a matrix or a vector X.

In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature. [ABSTRACT FROM AUTHOR]