Tutte introduced a V by V skew-symmetric matrix T = (tij), called the Tutte matrix, associated with a simple graph G= (V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai–Edmonds decomposition, we de...

This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting −1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or −1. The second remark...

Orthogonal systems in L2(ℝ), once implemented spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the skew-symmetric, tridiagonal, irreducible, have been recently fully characterised. In this paper we go step further, imposing extra requirement fast computation: specifically, that first N coefficie...

In reality, the process observations are away from the assumption that are normal distributed. The observations could be skew distributions which should use an asymmetric chart rather than symmetric chart. Consequently, this research aim to study the robustness of the asymmetric Tukey’s control chart for skew and non-skew distributions as Lognormal and Laplace distributions. Furthermore, the pe...

it is proved that by using bounds of eigenvalues of an interval matrix, someconditions for checking positive deniteness and stability of interval matricescan be presented. these conditions have been proved previously with variousmethods and now we provide some new proofs for them with a unity method.furthermore we introduce a new necessary and sucient condition for checkingstability of interv...

We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices (B1, . . . , Bm) and (C1, . . . , Cm), whether there exists an invertible matrix A such that for every i ∈ {1, . . . ,m}, ABiA = Ci. We show that this problem can be solved in randomized polynomial time over finite fields of o...

it is well known that the matrix exponential function has practical applications in engineering and applied sciences. in this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $x$-form. for instance, $e^{mathbf{a}t}$, $t^{mathbf{a}}$ and $a^{mathbf{a}t}$ will be evaluated by the new formulas in this par...

In recent years, the skew-normal models introduced by Azzalini (1985)—and their multivariate generalizations from Azzalini and Dalla Valle (1996)—have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usu...

A discrete gyroscopic system is characterized by 2n first-order ordinary differential equations defined one symmetric and skew-symmetric, which describes the motion of a spinning body containing elastic parts. In this paper, we consider inverse problems such system: Given partial spectral data, find that it desired data. The general solution problem given best approximation to pair matrices pro...