A general mathematical formulation of the n × n proper Orthogonal matrix, that corresponds to a rigid rotation in n-dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (n−2) principal axes. The latter, however, can be more conveniently replaced by only 2 Orthogonal directions that identify the plane of rotation...

where τ0 = 1. The τ -functions τn are given by the pfaffians of certain skew-symmetric matrix. We identify one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by 4N × 4N skew-symmetric constant matrix which we call the B-matrices. We then find that one can have M -soli...

In this note, we study the notion of structured pseudospectra. We prove that for Toeplitz, circulant and symmetric structures, the structured pseudospectrum equals the unstructured pseudospectrum. We show that this is false for Hermitian and skew-Hermitian structures. We generalize the result to pseudospectra of matrix polynomials. Indeed, we prove that the structured pseudospectrum equals the ...

In this work we propose a general framework for the structured perturbation analysis of several classes of structured matrix polynomials in homogeneous form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate e...

Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

We study the problem of feedback control for skew-symmetric and skewHamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke, et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, giv...

In this note, we consider the problem of computing the exponential of a real matrix. It is shown that if A is a real n × n matrix and A can be diagonalized over C, then there is a formula for computing e involving only real matrices. When A is a skew symmetric matrix, the formula reduces to the generalization of Rodrigues’s formula given in Gallier and Xu [1].

afael Navarro nstituto de Ciencia de Materiales de Aragón ICMA onsejo Superior de Investigaciones Científicas-Universidad de Zaragoza, Facultad de Ciencias laza San Francisco s/n 0 009 Zaragoza, Spain -mail: [email protected] Abstract. A theoretical framework to formulate and solve the problem of obtaining the objective refraction of an eye from aberrometric data is presented. Matrix formalism...