Let P = f" + (I ,,) , the direct sum of the p x p identity matrix and the negative of the q x q ident ity matrix. The fo llowing theorem is proved. TH EOHEM: If X = cZ where Z is a 4 x 4 P-orthogonal , P-skew-symmetric matrix and Ie I .;;; 2, there exist matrices A an.d B, both of which are P-orthogollal and P-skew-symmetric, sach that X = AB BA. Methods for o btaining certain matrices which sa...

Given an undamped gyroscopic system GðλÞ 1⁄4 Mλ þ CλþK with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX;TÞ and a skew-symmetric parameter matrix S . When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inv...

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric different...

An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, $A^T=A^{-1}$, respectively. We give necessary sufficient conditions on a complex so that it sum of type ``"orthogonal $+$ symmetric" in terms the Jordan form $A-A^T$. also "orthogonal skew-symmetric" $A+A^T$.

The sensitivity of a multiple eigenvalue of a matrix under perturbations can be measured by its Hölder condition number. Various extensions of this concept are considered. A meaningful notion of structured Hölder condition numbers is introduced and it is shown that many existing results on structured condition numbers for simple eigenvalues carry over to multiple eigenvalues. The structures inv...

The present investigation deals with the propagation of waves in a micropolar transversely isotropic layer. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and microrotation were also obtained. Finally, the numerical solution was carried out for aluminium epoxy material and the dispersion curv...

 In this paper we consider a general setting of skew-symmetric distribution which was constructed by Azzalini (1985), and its properties are presented. A suitable empirical estimator for a skew-symmetric distribution is proposed. In data analysis, by comparing this empirical model with the estimated skew-normal distribution, we show that the proposed empirical model has a better fit in den...

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×...

Abstract. By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli type matrices of a skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This gives an affirmative answer to an open pro...

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admit unfoldings to skew-symmetric matrices. We develop a combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in [1]. As a corollary, we use this algorithm to determine ...