Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds

In this paper, we investigate the Dirichlet problem for one type of vortex equations, which generalize the well-known Hermitian Yang-Mills equations, over general Hermitian manifolds.     

Generalized Hermitian codes

We investigate one-point algebraic geometry codes defined from curves related to the Hermitian curve. We obtain codes attaining new records on the parameters.     

Hermitian Veronesean Schemes

We introduce the idea of a Hermitian scheme and study some geometric properties. Examples and generalizations are given.     

Quaternion Hermitian Planes

The quaternion hermitian planes are defined, and are characterized by certain groups of automorphisms. For this purpose, characterizations of locally compact connected translation planes (in the context of stable planes) and compact connected projective desarguesian planes are given.     

Twistorial examples of almost Hermitian manifolds with Hermitian Ricci tensor

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold.     

On the intersection of Hermitian curves and of Hermitian surfaces

Kestenband [Unital intersections in finite projective planes, Geom. Dedicata 11(1) (1981) 107–117; Degenerate unital intersections in finite projective planes, Geom. Dedicata 13(1) (1982) 101–106] determines the structure of the intersection of two Hermitian curves of PG(2,q²), degenerate or not. In this paper we give a new proof of Kestenband's results. Giuzzi [Hermitian varieties over finite field, Ph.D. Thesis, University of Sussex, 2001] determines the structure of the intersection of two non-degenerate Hermitian surfaces and of PG(3,q²) when the Hermitian pencil defined by and contains at least one degenerate Hermitian surface. We give a new proof of Giuzzi's results and we obtain some new results in the open case when all the Hermitian surfaces of the Hermitian pencil are non-degenerate.     

Hermite矩阵方程

本文讨论矩阵方程X^*AX=A的求解,其中A为Hermite矩阵,X^*为X的转置矩阵.文中给出解的表示式.     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Differential operators for Hermitian Jacobi forms and Hermitian modular forms

Kim (Arch Math (Basel) 79(3):208–215, 2002) constructs multilinear differential operators for Hermitian Jacobi forms and Hermitian modular forms. However, her work relies on incorrect actions of differential operators on spaces of Hermitian Jacobi forms and Hermitian modular forms. In particular, her results are incorrect if the underlying field is the Gaussian number field. We consider more general spaces of Hermitian Jacobi forms and Hermitian modular forms over \(\mathbb {Q}(i)\), which allow us to correct the corresponding results in Kim (2002).     

反厄米特型Toeplitz线性方程组的反厄米特循环预处理子

本文主要研究了带位移的反厄米特型Toeplitz线性方程组Anx=b的一个新的反厄米特循环预处理子Cn,其中矩阵An的元素是函数f（θ）=a0＋ig（θ）的傅里叶系数．如果g（θ）是Wiener类实值函数,则矩阵Cn非奇异;且当n足够大时,矩阵（Cn^-1An）·（Cn^-1An）的谱以1为聚点,数值实验进一步显示了我们的预处理子是有效的．     

Compact Hermitian surfaces of Einstein type with respect to the Hermitian connection

Two Einstein-type conditions for the Hermitian curvature tensor are considered on a compact Hermitian surface: It is proved that if the symmetric part of the Ricci tensors is a scalar multiple of the metric with a negative constant, then the metric is Kaehler. If the Hermitian surface satisfies the Hermite-Einstein condition with a non positive constant, then the metric is Kaehler.Supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria.Supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

Hermitian varieties as codewords

We show that the incidence vector of any hermitian variety in the projective geometry PG _m–1(F _q ²), where q = p ^t , and p is a prime, is in the code over F _p of the symmetric design of points and hyperplanes of the geometry by using the theorem of Delsarte [8] that identifies this code with a nonprimitive generalized Reed-Muller code.     

The Hermitian Class Group

In this note, we study the Hermitian class group of orders over a Krulldomain and derive some exact sequences, permitting to calculate it inconcrete situations.In particular, we will show the Hermitian class group of a divisorialAzumaya algebra to be independent of its involution.