We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermit-ian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov–Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.

We first consider the following inverse eigenvalue problem: givenX ∈ Cn×m and a diagonal matrix Λ ∈ Cm×m, find n×nHermite-HamiltonmatricesK andM such thatKX MXΛ. We then consider an optimal approximation problem: given n × n Hermitian matrices Ka and Ma, find a solution K,M of the above inverse problem such that ‖K−Ka‖ ‖M−Ma‖ min. By using the MoorePenrose generalized inverse and the singular v...

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian (PS) splitting, and then establish a class of positivedefinite and skew-Hermitian splitting (PSS) methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS...

In an earlier paper [GKO95] we exploited the displacement structure of Cauchy-like matrices to derive for them a fast O(n) implementation of Gaussian elimination with partial pivoting. One application is to the rapid and numerically accurate solution of linear systems with Toeplitzlike coe cient matrices, based on the fact that the latter can be transformed into Cauchy-like matrices by using th...

The local and non-local vector Non-linear Schrodinger Equation (NLSE) with a general cubic non-linearity are considered in presence of linear term characterized, general, by non-hermitian matrix which under certain condition incorporates balanced loss gain coupling between the complex fields governing non-linear equations. It is shown that systems posses Lax pair an infinite number conserved qu...

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

We give some sufficient and necessary conditions for an element in a ring with involution to be Hermitian by using certain equations admitting solutions definite set the general solution representation.

This lecture takes a closer look at Hermitian matrices and at their eigenvalues. After a few generalities about Hermitian matrices, we prove a minimax and maximin characterization of their eigenvalues, known as Courant–Fischer theorem. We then derive some consequences of this characterization, such as Weyl theorem for the sum of two Hermitian matrices, an interlacing theorem for the sum of two ...

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly timedependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator p...