A computationally e cient method for structured covariance matrix estimation is presented. The proposed method provides an Asymptotic (for large samples) Maximum Likelihood estimate of a structured covariance matrix and is referred to as AML. A closed-form formula for estimating Hermitian Toeplitz covariance matrices is derived which makes AML computationally much simpler than most existing Her...

Let V be 2n-dimensional vector space over a field K equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let K0 ⊆ K be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, . . . , 2n}, there exist natural representations of the groups G ∼= U(2n,K/K0) and H = G∩SL(V ) ∼= SU(2n,K/K0) on the k-th exterior power of V . With the aid of linear...

Among P-pseudo-Hermitian Hamiltonians H = P−1H† P with real spectra, the “weakly pseudo-Hermitian” ones (i.e., those employing non-self-adjoint P 6 = P†) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever P 6 = P†, the current involutive operator of charge C gets complemented by a nonequivalent alternative involutive quasip...

Comparing the lopsided Hermitian/skew-Hermitian splitting (LHSS) method and Hermitian/skewHermitian splitting (HSS) method, a new criterion for choosing the above two methods is presented, which is better than that of Li, Huang and Liu [Modified Hermitian and skew-Hermitian splitting methods for nonHermitian positive-definite linear systems, Numer. Lin. Alg. Appl., 14 (2007): 217-235]. Key-Word...

The unitary group of the hyperbolic hermitian space of dimension two over a quaternion division algebra over a number field is a non-quasisplit inner form of Sp(4), and does not have a parabolic subgroup corresponding to the Klingen parabolic subgroup. However, it has CAP representations with respect to the Klingen parabolic subgroup. We construct them by using the theta lifting from the unitar...

Consider a bipartite quantum system consisting of two subsystems A and B, of dimensions nA and nB , respectively. The overall system has dimension n := nAnB. The state of the total system is represented by an Hermitian n×nmatrix ρ, called the density matrix, which is positive semi-definite and has trace equal to one. Special types of density matrices are product matrices, i.e., matrices ρprod o...

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A+XTX∗, originally studied in Marčenko and Pastur [4], is presented. Here, X (N×n), T (n×n), and A (N×N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N → c > 0 as N → ∞. Under addtional ass...

We list some reasons why among P−pseudo-Hermitian Hamiltonians H = P HP with real spectra, the “weakly pseudo-Hermitian” ones (i.e., those employing P 6 = P, even when belonging to the former class) form such an interesting special family that they deserve a special name and attention. In particular we show that for P 6 = P, the current auxiliary involutive operator of charge C gets complemente...

We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra.

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspacemethods.We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding preconditi...