This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...

We study the Hermitian positive definite solutions of the nonlinear matrix equation X A∗X−2A I, where A is an n × n nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of X A∗X−2A I are presented ...

Abstract. Consider an N × N hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N → ∞, the same as that of hermitian random matrices from GUE. We prove this con...

In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermi...

In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). Now we try to release the hermitian restriction on observables. This is only a mathematical interest. In this case we give several non-hermitian ex...

We demonstrate that the recent paper by Jana and Roy entitled “Non-Hermitian quantum mechanics with minimal length uncertainty” [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Her...

It has been a central problem in the theory of quadratic forms to find integers represented by quadratic forms. The celebrated Four Square Theorem by Lagrange [10] was an outstanding result in this study. Ramanujan generalized this theorem and found 54 positive definite quaternary quadratic forms which represent all positive integers [13]. We call a positive definite quadratic form universal, i...

By applying the higher order Darboux algorithm to an exactly solvable non Hermitian PT symmetric potential, we obtain a hierarchy of new exactly solvable non Hermitian PT symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudo supersymmetry. We also show that this formalism can be used to generate a la...

This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...