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.     

Generalized canonical factorization of matrix and operator functions with definite hermitian part

It is proved that rational matrix functions with definite hermitian part on the real line admit a generalized canonical factorization. The functions are allowed to have poles on the real line. A generalization of this result to a class of operator functions is obtained as well.Partially supported by an NSF grant     

A note on backward perturbations for the hermitian eigenvalue problem

Through different orthogonal decompositions of computed eigenvectors we can define different Hermitian backward perturbations for a Hermitian eigenvalue problem. Certain optimal Hermitian backward perturbations are studied. The results show that not all the optimal Hermitian backward perturbations are small when the computed eigenvectors have a small residual and are close to orthonormal.Dedicated to Åke Björck on the occasion of his 60th birthdayThis work was supported by the Swedish Natural Science Research Council under Contract F-FU 6952-302 and the Department of Computing Science, Umeå University.     

Mayer-vietoris sequences for the hermitian picard group

In this note, we study the functorial behaviour of the hermitian Picard group with respect to centrally induced ring morphisms. As an application, we construct Mayer-Vietoris sequences which provide an efficient tool to concretely calculate the hermitian Picard group.     

二次矩阵方程AX~2+BX+C=0的可对角化解

给出了二次矩阵方程AX2+BX+C=0的特征值和特征子空间的定义,然后运用其特征子空间的维数或特征向量刻画了该二次矩阵方程存在可对角化解的充要条件.     

On the u-invariant of hermitian forms

Let K be a field of characteristic not 2 and A a central simple algebra with an involution σ. A result of Mahmoudi provides an upper bound for the u-invariants of hermitian forms and skew-hermitian forms over (A, σ) in terms of the u-invariant of K. In this paper we give a different upper bound when A is a tensor product of quaternion algebras and σ is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.     

A supplement to oscillation and comparison theory for hermitian differential systems

For hermitian differential systems there is presented a characterization of a conjoined family of solutions (U; V) for which U is nonsingular between a pair of mutually conjugate points, or on an interval contiguous to a focal point. The general results are obtained without any assumption of normality of the system. For a class of systems which are direct generalizations of the canonical form of nonsingular accessory systems associated with a simple integral variational problem with no differential equation restraints, there is established a special comparison theorem involving a pointwise monotoneity property of the related integrand functions. The final section is devoted to a special result for systems consisting of a pair of scalar equations with real coefficients, which in case the coefficient functions are continuous provides the result of the basic theorem of a recent paper [Applicable Analysis 2 (1972), 355–376] by J. B. Diaz and J. R. McLaughlin.     

Inertia possibilities for completions of partial hermitian matrices

A partial Hermitian matrix is one in which some entries are specified and others are considered to be free (complex) variables. Assuming the undirected graph of the specified entries is chordal, it is shown that, with certain mild restrictions, a partial Hermitian matrix may be completed to a Hermitian matrix with any inertia allowed by the specified principal submatrices through the interlacing inequalities. This generalizes earlier work dealing with the existence of positive definite completions, and. as before, the chordality assumption is, in general, necessary. Further related observations dealing with Toeplitz completions and the minimum eigenvalues of completions are also made, and these raise additional questions.     

A local–global principle for algebras with involution and hermitian forms

 Weakly hyperbolic involutions are introduced and a proof is given of the following local–global principle: a central simple algebra with involution of any kind is weakly hyperbolic if and only if its signature is zero for all orderings of the ground field. Also, the order of a weakly hyperbolic algebra with involution is a power of two, this being a direct consequence of a result of Scharlau. As a corollary an analogue of Pfister's local–global principle is obtained for the Witt group of hermitian forms over an algebra with involution. Received: 29 October 2001; in final form: 9 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 16K20, 11E39     

A local hermitian RBF meshless numerical method for the solution of multi‐zone problems

The local Hermitian interpolation (LHI) method is a strong‐form meshless numerical technique in which the solution domain is covered by a series of small and heavily overlapping radial basis function (RBF) interpolation systems. Aside from its meshless nature and the ability to work on very large scattered datasets, the main strength of the LHI method lies in the formation of local interpolations, which themselves satisfy both boundary and governing PDE operators, leading to an accurate and stable reconstruction of partial derivatives without the need for artificial upwinding or adaptive stencil selection. In this work, an extension is proposed to the LHI formulation which allows the accurate capture of solution profiles across discontinuities in governing equation parameters. Continuity of solution value and mass flux is enforced between otherwise disconnected interpolation systems, at the location of the discontinuity. In contrast to other local meshless methods, due to the robustness of the Hermite RBF formulation, it is possible to impose both matching conditions simultaneously at the interface nodes. The procedure is demonstrated for 1D and 3D convection–diffusion problems, both steady and unsteady, with discontinuities in various PDE properties. The analytical solution profiles for these problems, which experience discontinuities in their first derivatives, are replicated to a high degree of accuracy. The technique has been developed as a tool for solving flow and transport problems around geological layers, as experienced in groundwater flow problems. The accuracy of the captured solution profiles, in scenarios where the local convective velocities exceed those typically encountered in such Darcy flow problems, suggests that the technique is indeed suitable for modeling discontinuities in porous media properties. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1201–1230, 2011     

Extreme rays for the large cone of hermitian generalized matrix functions

Given a <artwork name="GLMA31007ei1">-valued function f with domain <artwork name="GLMA31007ei2">, the symmetric group on {1,2,…, m}, we define the generalized matrix function [ f ](?), or d_f (?), in the usual way on the set of all m× m complex matrices. Letting <artwork name="GLMA31007ei3"> denote the set of all m× m positive semi-definite Hermitian matrices we consider the cone K _m whose elements are the Hermitian functions <artwork name="GLMA31007ei4"> such that [ f ]( A)≥0 for all <artwork name="GLMA31007ei5">. The extreme rays in K _m are fundamental to an understanding of the linear inequalities that result by restricting the generalized matrix functions [ f ](?) to the sets <artwork name="GLMA31007ei6">. In particular, the resolution of Lieb's permanent dominance conjecture, and certain similar conjectures such as the conjecture of Soules, will likely require identification and careful analysis of these rays. Grone, Merris, and Watkins have shown that the determinant function det(?), which is [ f ](?) if f is the signum function, is extreme in K _m for each m. We identify additional rays that are extreme for all m. In particular, we associate with each 2-term partition <artwork name="GLMA31007ei7"> of {1,2,…, m} an element <artwork name="GLMA31007ei8"> that is shown to be extreme in K _m for each m. If <artwork name="GLMA31007ei9"> is trivial, then <artwork name="GLMA31007ei10"> reduces to the determinant function; hence, our results are a natural extension of the result of Grone, Merris, and Watkins. Moreover <artwork name="GLMA31007ei11">, like det ( A), is expressible as a function of the eigenvalues of certain matrices related to A. Additional classes of extreme rays are also presented.