This paper is concerned with the Hermitian definite generalized eigenvalue problem A− λB for block diagonal matrices A 1⁄4 diagðA11; A22Þ and B 1⁄4 diagðB11; B22Þ. Both A and B are Hermitian, and B is positive definite. Bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices are established. These bounds are generally of linear order with respect to the perturbations...

We study the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. present purely linear algebra-based approach derive explicit computable formulas nearest $(A-\Delta_A)+s(E-\Delta_E)$ su...

Non-Hermitian systems with parity-time reversal ($\mathcal{PT}$) or anti-$\mathcal{PT}$ symmetry have attracted a wide range of interest owing to their unique characteristics and counterintuitive phenomena. One the most extraordinary features is presence an exceptional point, across which phase transition spontaneously broken $\mathcal{PT}$ takes place. We implement Floquet Hamiltonian single q...

A non-Hermitian PφTφ-symmetrized spherically-separable Dirac Hamiltonian is considered. It is observed that the descendant Hamiltonians Hr, Hθ, and Hφ play essential roles and offer some ”user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a PφTφ-symmetrized Hφ, we have shown that the conventional relativistic energy eigenvalues are recoverable. We h...

In this paper, we further investigate the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. When A is non-symmetric positive definite, the convergence conditions are obtained, which generalize some results of Jiang and Cao [M.-Q. Jiang and Y. Ca...

Finite dimensional indefinite inner product spaces of vector valued rational functions which are (1) invariant under the generalized backward shift and (2) subject to a structural identity, and subspaces and “superspaces” thereof are studied. The theory of these spaces is then applied to deduce a generalization of a pair of rules due to Iohvidov for evaluating the inertia of certain subblocks o...

A bstract The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation mixed-state entanglement. However, so far, it has only been studied for matrices corresponding to subsystems globally pure states. Here, we consider genuine example mixed state one-dimensional massless Dirac fermions in system at finite temperature an...

We propose construction of a unique and definite metric (η +), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out.

In this paper, we first consider the inverse eigenvalue problem as follows: Find a matrix A with specified eigen-pairs, where A is a Hermitian and generalized skewHamiltonian matrix. The sufficient and necessary conditions are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by LS . Then the best approximation problem for the inverse eigen...

We prove in this note the convexity of the functions u ◦ λ and more generally u ◦ λB on the space of Hermitian matrices, for B a fixed positive definite hermitian matrix, when u : R → R ∪ {+∞} is a symmetric convex function which is lower semi-continuous on R, and finite in at least one point of R. This is performed by using some optimisation techniques and a generalized Ky Fan inequality. To c...