Consider the matrix equation AXA∗ + BY B∗ = C. A matrix pair (X0, Y0) is called a Hermitian nonnegative-definite solution to the matrix equation if X0 and Y0 are Hermitian nonnegative-definite and satisfy AX0A∗ + BY0B∗ = C. We give necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation, and further derive a representation of the...

The neighbour method of Kneser can be adapted to the hermitian case. Generalizing results of Hof91], we show that it can be used to classify any genus in a hermitian space of dimension 2 by neighbour steps at suitable primes. The method was implemented for positive deenite hermitian lattices (not necessarily free) over Q(p d). A table of class numbers of unimodular genera and the largest minima...

in this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring r such that, every regular matrix admits a diagonal power-reduction.

We study a non-Hermitian generalization of quantum systems in which an imaginary vector potential is added to the momentum operator. In the tight-binding approximation, we make the hopping energy asymmetric in the Hermitian Hamiltonian. In a previous article, we conjectured that the non-Hermitian critical point where the energy gap vanishes is equal to the inverse correlation length of the Herm...

We consider the description of open quantum systems with probability sinks (or sources) in terms of general non-Hermitian Hamiltonians. Within such a framework, we study novel possible definitions of the quantum linear entropy as an indicator of the flow of information during the dynamics. Such linear entropy functionals are necessary in the case of a partially Wigner-transformed non-Hermitian ...

where S is an n × n positive definite Hermitian matrix, dS is equivalent to the dH in (1) restricted in the subset of positive definite Hermitian matrices, and E+ is a subset of R+. The integral H(E) is in some sense the limiting case of L(E+), since H(E) can be evaluated by Hermitian polynomials, while L(E+) by Laguerre polynomials in the same way [8]. It is a classical result that Hermitian p...

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant ⋆-scalar curvature which satisfies the Kaehler condition at the point in question.

Abstract. In this paper, some new lower bounds for the smallest singular value of a square complex matrix A are derived. A key tool to obtain these bounds is using some Hermitian matrices which are, in the sense of Loewner matrix ordering, below the Hermitian part of A or, more generally, below the Hermitian part of the unitary equivalences of A. Two types of new bounds are proposed. The first ...

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA +BY B = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y =...