Polynomial equations over octonion algebras

In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].

Localization theorems for matrices and bounds for the zeros of polynomials over quaternion division algebra

In this paper, we derive Ostrowski and Brauer type theorems for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic matrix. After that, a sufficient condition for the stability of a quaternionic matrix is given that generalizes the stability condition for a complex matrix. Finally, a characterization of bounds is derived for the zeros of quaternionic polynomials.

An inverse eigenproblem for generalized reflexive matrices with normal $k+1$-potencies

Let $P,~Q\in\mathbb{C}^{n\times n}$ be two normal $\{k+1\}$-potent matrices, i.e., $PP^{*}=P^{*}P,~P^{k+1}=P$, $QQ^{*}=Q^{*}Q,~Q^{k+1}=Q$, $k\in\mathbb{N}$. A matrix $A\in\mathbb{C}^{n\times n}$ is referred to as generalized reflexive with two normal $\{k+1\}$-potent matrices $P$ and $Q$ if and only if $A=PAQ$. The set of all $n\times n$ generalized reflexive matrices which rely on the matrices $P$ and $Q$ is denoted by $\mathcal{GR}^{n\times n}(P,Q)$. The left and right inverse eigenproblem of such matrices ask from us to find a matrix $A\in\mathcal{GR}^{n\times n}(P,Q)$ containing a given part of left and right eigenvalues and corresponding left and right eigenvectors. In this paper, first necessary and sufficient conditions such that the problem is solvable are obtained. A general representation of the solution is presented. Then an expression of the solution for the optimal Frobenius norm approximation problem is exploited. A stability analysis of the optimal approximate solution, which has scarcely been considered in existing literature, is also developed.

Simulation of Vibroacoustic Problem Using Coupled FE/FE Formulation and Modal Analysis

Vibroacoustic consists of the interaction between elastic and acoustic waves. This interaction causes the acoustic pressure to exert a force to the structure and the structural motion produces an effective fluid load. Hence, studying vibroacoustic problems requires that both elastic structure and fluid to be modeled. In this paper, a simple vibroacoustic problem is modeled by a coupled FE/FE modal analysis method in which a lumped mass representation is considered. The numerical results are represented for an elastic plate backed cavity. In this method, the coupling system is projected on some structural modes in vacuum and some rigid cavity modes. The obtained system is put in a diagonal form by solving left and right eigenvalues problem. Some results are presented to illustrate the effect of the modal analysis and the lumped mass formulation. Some numerical results were compared to analytical solutions.