Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

矩阵Hadamard乘积的不等式

本文运用矩阵 Hadamard乘积及控制不等式的性质 ,获得了若干 Hermite及斜 Hermite矩阵特征值的不等式     

Generalized minimax and interlacing theorems

We give some steps towards a unified theory of Courant-Fischer minimax-type formulas and Cauchy interlacing-type inequalities that have been obtained for the eigenvalues of Hermitian matrices, for singular values of complex matrices, and for invariant factors of integral matrices

We also unify and extend work on eigenvalues, singular values, and invariant factors of pairs of matrices and their sum or product     

Inertial properties of eigenvalues,II

Inequalities involving the eigenvalues of conjunctive Hermitian matrices are established, and shown to contain a recent result of Machover, the law of inertia, and the interlacing inequalities.     

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 49–59. Translated by L. Yu. Kolotilina.     

Equalities and inequalities for inertias of hermitian matrices with applications

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 2×2 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 A-BXB^∗ 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 AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Eigenvalue Inequalities and Schubert Calculus

Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A₁,…, A_r ? ¢^n×n with the spectrum of the sum A₁ + … + A_r. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.     

Weyl and Lidski? Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidskiǐ inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

Weyl and Lidski(i) Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidski(i) inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

Eigenvalue inequalities associated with the cartesian decomposition

Let T=A+iB where AB are Hermitian matrices. We obtain several inequalities relating the l_p distance between the eigenvalues of A and those of iB with the Schatten p-norm of T. The majorization results which lead to these inequalities are also used to get simple proofs of some known lower and upper bounds for the determinant of T.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

Trace inequalities involving Hermitian matrices

Some trace inequalities for Hermitian matrices and matrix products involving Hermitian matrices are presented.     

Concentration of Haar measures, with an application to random matrices

We show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective Haar measures. As an application, we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random Hermitian matrices, with possible applications in free probability. The advantage over existing techniques is that the new method can deal with functions that are non-Lipschitz or even discontinuous with respect to the usual metrics.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

The strengthened versions of the additive and multiplicative Weyl inequalities

The strengthened versions of the classical additive and multiplicative Weyl inequalities for the singular values of A + B and AB*, where A and B are rectangular matrices, and for the eigenvalues of A + B and AB, where A and B are Hermitian matrices, are established under certain assumptions on the subspaces spanned by some singular vectors or eigenvectors, respectively, of A and B. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 39–59.     

Rearrangement inequalities for Hermitian matrices

Hermitian matrices can be thought of as generalizations of real numbers. Many matrix inequalities, especially for Hermitian matrices, are derived from their scalar counterparts. In this paper, the Hardy-Littlewood-Pólya rearrangement inequality is extended to Hermitian matrices with respect to determinant, trace, Kronecker product, and Hadamard product.     

Remarks on eigenvalues and eigenvectors of Hermitian matrices,berry phase,adiabatic connections and quantum Hall effect

The singularities of the stratification of the space of the Hermitian matrices according to the multiplicities of the eigenvalues are described as an informal complexification of the previous study of the space of the real symmetric matrices. The degeneration of the spectral sequence associated to this stratification provides some strange combinatorial identities. The eigenvector bundles over the manifold of the Hermitian matrices with simple spectra are equiped with the natural connections, describing also the adiabatic approximation to the oscillations of the linear systems defined by the slowly varying skew Hermitian matrices. The curvature of this connection is singular at the codimension 3 variety of the Hermitian matrices having multiple eigenvalues. The resulting jumps of the integrals of the curvature form at the crossings of this variety by the moving surface of integration are responsible for the quantum Hall effect.This article is an expanded version of a lecture read 25 March, 1994, at the conference on Real Algebraic Geometry at the Institut Henry Poincaré, Paris     

A matrix version of the Ky Fan generalization of the kantorovich inequality

Ky Fan extended some inequalities for Hermitian positive definite matrices to sums of such matrics. These and similar results are further generalized by showing that the inequalities are maintained with orthogonal matrices substituted for vectors.     

半正定Hermitian矩阵的广义Schur补的Lner偏序和特征值

首先得到了半正定 Hermitian矩阵的方幂的广义 Schur补的 L owner偏序的一些结果 ,然后改进了半正定 Hermitian矩阵的 Schur补的交错不等式 .