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Let Ai,P, i=1,…,m, be n by n complex matrices such that each Ai is positive definite. It is shown, among other inequalities, that If 0

In this paper, to find the numerical solution of the linear systems with the non-Hermitian positive definite matrix, a preconditioned two-sweep shift splitting (PTSS) iteration method is introduced...

We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system JS of all symmetric n by n(n≥2) matrices over a fiel...

The Hermitian and skew-Hermitian splitting (HSS) method is a well-known method employed in solving the non-Hermitian positive definite linear systems. However, the performance of the HSS method hig...

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix ...

We consider several special functions defined by matrix integrals on Hermitian matrix space, namely, analogues of Gauss hypergeomeric, Kummer's confluent hypergeometric, Bessel, Hermite-Weber and A...

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...

The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation A ∗ N X = B under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian sp...

In this paper, we consider a linear constraint problem of Hermitian unitary symplectic matrices and its approximation. By constructing a simple unitary matrix U, we verify that Hermitian unitary sy...

This paper is concerned with stability of deficiency indices of Hermitian subspaces (i.e. linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invarianc...

In this article, we introduce a new matrix associated with a multidigraph, named as the complex adjacency matrix. We study the spectral properties of bipartite multidigraphs corresponding to the complex adjacency matrix. It is well known that a simple undirected graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric about the origin (with multiplicity). We show that the result is not true in general for multidigraphs and supply a class of non-bipartite multidigraphs which have this property. We describe the complete spectrum of a multi-directed tree in terms of the spectrum of the corresponding modular tree. As a consequence, we get a class of Hermitian matrices for which the spectrum of a matrix in the class and the spectrum of the modulus (entrywise) of the matrix are the same.

Given an n-vertex graph G with maximum degree Δ, the mixed graph DG is constructed from G by orienting some of its edges, where G is called the underlying graph of DG. Let rH(DG) be the H-rank of D...

We provide a concise overview of some iterative approaches, by which large and sparse eigenvalue problems are successfully dealt with. Our main goal in this paper is to prove the global convergence...

In this paper, necessary and sufficient conditions for the existence of the Hermitian positive definite solution for a nonlinear matrix equation are derived. Then, a sufficient condition for the ex...

We study the class of bihomogeneous polynomials r(z,z¯) on Cn×Cn for which there is a positive integer d such that r(z,z¯)z2d can be written as a Hermitian sum of squares. We reinterpret th...

In this paper, we construct a family of vertex operator algebras VJ,r parameterized by a non-zero complex number r, whose Griess algebras V2 are isomorphic to Hermitian Jordan algebras J of type C ...

In a previous paper the authors developed an algorithm to classify certain quaternary quadratic lattices over totally real fields. The present article applies this algorithm to the classification o...

In this paper, the following problem is considered: given two Hermitian matrices H and K and two real numbers x and y, determine a positive semidefinite matrix ρ such that the von Neumann entropy −...

The quantity ⟨ψ(t−Δt/2)|aˆ|ψ(t+Δt/2)⟩ given for a square integrable wavefunction ψ allows one to define very different kinds of measurables, such as absorption spectrum (as a Fourier transform (FT)...

Let R be a unital ∗-ring and let a,e,f∈R with e,f invertible Hermitian elements. In this paper, we define two types of outer generalized inverses, called pseudo e-core inverses and pseudo f-dual co...

For solving ill-posed inverse problems, we establish a modified version of special Hermitian and skew-Hermitian splitting (SHSS) iteration method, called the modified SHSS (MSHSS) method in this pa...

A mixed graph is obtained from an unoriented graph by orienting a subset of its edges. Yu, Liu, and Qu in 2017 have established the expression for the determinant of Hermitian (quasi-) Laplacian matrix of a mixed graph. Here we find general expressions for all minors of Hermitian (quasi-) Laplacian matrix of mixed graphs., 10 pages

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the norm $\nu$; (3) We make a precise connection with the Cayley-Dickson process for structurable algebras.

Quantum-statistical properties of the phase of electromagnetic field in the case of phase superpositions of coherent states of the type Nα,β,ξneiϕα+eiξneiϕβ have been studied. Formulas for the aver...

Given a graph G, the mixed graph is obtained from G by orienting some of its edges (G is also called the underlying graph of ). Let be the Hermitian energy of and let be the matching number...

This paper aims to consider the Hermitian solutions of reduced biquaternion matrix equation ( A X B , C X D ) = ( E , G ) , where X is an unknown reduced biquaternion Hermitian matrix, and A, B, C,...

This paper deals with hermitian operators and surjective linear isometries, between spaces of Lipschitz maps, defined on a compact metric space, with values in a finite dimensional vector space. Th...

Let Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i be a linear map on the Kronecker product of spaces of Hermitian matrices H n i of size n i ≥ 3 . (If d=1, we identify ⨂ i = 1 d H n i with H n 1 .) We esta...

We investigate the genera of quotient curves ℋq∕G of the 𝔽q2-maximal Hermitian curve ℋq, where G is contained in the maximal subgroup ℳq≤Aut(ℋq) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated with ℋq. To this aim, a geometric and group-theoretical description of ℳq is given. The genera of some other quotients ℋq∕G with G≰ℳq are also computed. In this way we obtain new values in the spectrum of genera of 𝔽q2-maximal curves. The complete list of genera g>1 of quotients of ℋq is given for q≤29, as well as the genera g of quotients of ℋq with g>q2−q+3032 for any q. As a direct application, we exhibit examples of 𝔽q2-maximal curves which are not Galois covered by ℋq when q is not a cube. Finally, a plane model for ℋq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Sz\`ekelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau., Comment: Final version. 14 pages. To appear in Complex Variables and Elliptic equations

We present three new results, each concerned with ranks of principal submatrices. The first connects the ranks of sums of rank-one matrices with that of certain associated partial sums. The other r...

In this paper, we first investigate the matrix equation in a new fashion, where A, B, C, D and G are arbitrary matrices. Then, we establish some necessary and sufficient conditions for the existence of a solution of , where A, B are arbitrary matrices and C is a Hermitian matrix. In the special case when , we determine the general solution of , where A is an arbitrary matrix and C is a Hermitian matrix.

In this paper we investigate the matrix equation . Some sufficient and necessary conditions for the existence of Hermitian positive definite solutions as well as for the existence of the smallest Hermitian positive definite solution of the considered equation are derived. Two iterative methods for computing the smallest positive definite solution are proposed. Some numerical examples are presented to illustrate the theoretical results.

Let $M$ be an arbitrary complex manifold and let $L$ be a Hermitian holomorphic line bundle over $M$. We introduce the Berezin-Toeplitz quantization of the open set of $M$ where the curvature on $L$ is non-degenerate. The quantum spaces are the spectral spaces corresponding to $[0,k^{-N}]$ ($N>1$ fixed), of the Kodaira Laplace operator acting on forms with values in tensor powers $L^k$. We establish the asymptotic expansion of associated Toeplitz operators and their composition as $k\to\infty$ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin-Toeplitz quantization for semi-positive and big line bundles., 44 pages; v.2 is a final update to agree with the published paper

A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.

In this paper, we establish some conditions for the existence and the representations for the Hermitian reflexive and Hermitian anti-reflexive, and nonnegative definite reflexive solutions to the matrix equation AX=B with respect to a generalized reflection matrix P by using the Moore-Penrose inverse. Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided.

In standard quantum theory, the Hamiltonian describing a physical system is assumed to be Hermitian in order to guarantee the energy spectrum to be real and the time evolution to be unitary. In recent years, it was recognized that non-Hermitian Hamiltonians with parity-time ($ \mathcal PT $) symmetry can exhibit entirely real spectra, raising the possibility for extending the quantum theory to complex domain and hence stimulated growing interest in recent years. Many proposals have been presented for realizing $ \mathcal PT $-symmetric Hamiltonians in various physical systems. Among them $ \mathcal PT $-symmetric coherent atomic gases are special and possess many unique advantages, including the possibility to obtain authentic $ \mathcal PT $-symmetric refractive indexes (which have balanced gain and loss in the whole space), the capability to actively control and precisely manipulate system parameters in situ, and the feasibility to acquire large Kerr nonlinearity based on the resonance character between light and atoms. In this article, we review various schemes for the realization of $ \mathcal PT $ symmetry with coherent atomic gases, elucidate their interesting properties and promising applications. In particular, the non-linear optical effect in the $ \mathcal PT $-symmetric atomic gases are described, which may be served as useful building blocks for developing novel photonic devices with active light control at very low power level.

Let R be a unital ring with involution, we give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a in R and n is an integer greater than or equal to 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0, a^n+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0, a^n+p is invertible. We also get a new characterization for both core invertible and dual core invertible of a regular element by units, and their expressions are shown. In particular, we prove that for n is an integer greater than or equal to 2, a is both Moore-Penrose invertible and group invertible if and only if (a*)^n is invertible along a., 18 pages

We determine reducibility points for a certain family of induced representations for quaternionic hermitian and anti-hermitian groups over a p-adic field. We do this via Bushnell and Kutzko’s method of types and covers.

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold it provides the Atiyah-Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations of the unitary group used are not irreducible, causing problems in considering invariance under a group larger than U(n). This motivates either the development of Clifford analysis over a complex vector space with respect to a Hermitian inner product or the development of Dirac-type operators on Cauchy-Riemann structures., 14 pages; submitted to Complex Variables and Elliptic Equations

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by the early introduction of complex arithmetic. Most elementary properties of complex numbers have immediate matrix analogues and many important theorems can be deduced, or at least postulated, from the basics of complex arithmetic alone.

A class of quaternion matrices called generalized -(anti-)bi-Hermitian matrices is defined which incorporates the -(anti-)bi-Hermitian matrices mentioned by Yuan et al. [Linear Multilinear Algebra. 63;2015:1849–1863] as special cases. In the earlier referred work, Yuan et al. have derived explicit formulas for the least-squares -(anti-)bi-Hermitian solutions of the coupled matrix equations. In this paper, an efficient iterative algorithm is proposed to numerically find the generalized least-squares -(anti-)bi-Hermitian solutions of the coupled matrix equationsThe validity and efficiency of the presented algorithm is examined by some test experiments.

Let F1 be an equivalence of type I between two categories of modules with hermitian forms. In other words, suppose F is an equivalence between the two underlying categories of modules, and F1 is an...

In this note, based on the previous work by Pour and Goughery (New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems Numer. Algor. 69 (2015) 207–225), we further discuss this new Hermitian and skew-Hermitian splitting (described as NHSS) methods for non-Hermitian positive definite linear systems. Some new convergence conditions of the NHSS method are obtained, which are superior to the results in the above paper.

A Buzano type inequality for two nonnegative Hermitian forms is obtained. Applications to inequalities for norm and numerical radius of bounded linear operators in complex Hilbert spaces are given.

In ‘A Hermitian analogue of the Morita theorems’ by Dasgupta and Dasgupta, we proved a Hermitian Morita II. There is an error in the proof of Proposition 3.1 which nullifies the paper.

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.

This article is concerned with the following problem: given , of which either A or B is Hermitian, and given , determine a vector such that with , referred to as a generalized Rayleigh quotient. Such a vector w is called a generating vector for z. Our approach builds on the fact that the field of values of a linear pencil can be accurately approximated under the compression into bidimensional linear pencils, in which case the problem has an exact solution. The efficiency of the presented algorithms is discussed and numerical examples are included.

For given skew-Hermitian unitary matrix J, i.e. J=−JH, JHJ=JJH=I, a matrix A∈Cn×n is termed generalized Hamiltonian matrix if JAJ=AH. In this paper, an iterative method is constructed to solve the generalized Hamiltonian solutions of the coupled Sylvester matrix equations with conjugate transpose. It is proved that the iterative method is unconditionally convergent for any initial generalized Hamiltonian matrices. With it, the generalized Hamiltonian solutions can be obtained within finite iteration steps in the absence of roundoff errors. Finally, numerical examples are presented to illustrate the efficiency and applicability of the method.