A matrix S ∈ C2m×2m is symplectic if SJS∗ = J , where J = [ 0 −Im Im 0 ] . Symplectic matrices play an important role in the analysis and numerical solution of matrix problems involving the indefinite inner product x∗(iJ)y. In this paper we provide several matrix factorizations related to symplectic matrices. We introduce a singular value-like decomposition B = QDS−1 for any real matrix B ∈ Rn×...

First, we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We also determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. Next, we analyze the effe...

The skew energy of an oriented graph was introduced by Adiga, Balakrishnan and So in 2010, as one of various generalizations of the energy of an undirected graph. Let G be a simple undirected graph, and Gσ be an oriented graph of G with the orientation σ and skew-adjacency matrix S(Gσ). The skew energy of the oriented graph Gσ , denoted by ES(G), is defined as the sum of the norms of all the ei...

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...

The time derivative of a rotation matrix equals the product of a skew-symmetric matrix and the rotation matrix itself. This article gives a brief tutorial on the well-known result.

let x1;x2;...;xn have a jointly multivariate exchangeable normal distribution. in this work we investigate another proof of the independence of x and s2 using order statistics. we also assume that (xi ; yi); i =1; 2;...; n; jointly distributed in bivariate normal and establish the independence of the mean and the variance of concomitants of order statistics.

We propose a two-level iterative scheme for solving general sparse linear systems. The proposed consists of preconditioner that increases the norm skew-symmetric part relative to rest and makes main diagonal coefficient matrix as close identity possible so preconditioned system is shifted possible. then solved via particular Minimal Residual Method Shifted Skew-Symmetric Systems (MRS). This lea...

In motion Kinematics, it is well-known that the time derivative of a 3 × 3 rotation matrix equals a skew-symmetric matrix multiplied by the rotation matrix where the skew symmetric matrix is a linear (matrix-valued) function of the angular velocity and the rotation matrix represents the rotating motion of a frame with respect to a reference frame. The equation is widely used in engineering, e.g...