Related papers: Stability of quaternion matrix polynomials
Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in…
Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with…
Sensitivity of an eigenvalue $\lambda_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $\kappa_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and…
In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…
We prove that in a theory $T$ stable over a predicate $P$, for any $\lambda > |T|$, there is a $\lambda$-prime model over any complete set A with a $\lambda$-saturated $P$-part.
Matrix polynomials with unitary/doubly stochastic coefficients form the subject matter of this manuscript. We prove that if $P(\lambda)$ is a quadratic matrix polynomial whose coefficients are either unitary matrices or doubly stochastic…
Assume that $\{a_{n};\,n\geq0\}$ is a sequence of positive numbers and $\sum a_{n}^{\,-1}<\infty$. Let $\alpha_{n}=ka_{n}$, $\beta_{n}=a_{n}+k^{2}a_{n-1}$ where $k\in(0,1)$ is a parameter, and let $\{P_{n}(x)\}$ be an orthonormal polynomial…
As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…
A polynomial $P \in \mathbb{C}[z_1, \ldots, z_d]$ is strongly $\mathbb{D}^d$-stable if $P$ has no zeroes in the closed unit polydisc $\overline{\mathbb{D}}^d.$ For such a polynomial define its spectral density function as…
In this paper, we prove the exponential stability property of a class of mechanical systems represented in the port-Hamiltonian framework. To this end, we propose a Lyapunov candidate function different from the Hamiltonian of the system.…
In theory of one complex variable, Gauss-Lucas Theorem states that the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the polynomial. The exact analogue of this result cannot hold, in general,…
The theory of polar forms of polynomials is used to provide for sharp bounds on the radius of the largest possible disc (absolute stability radius), and on the length of the largest possible real interval (parabolic stability radius), to be…
We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many…
This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx {{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance,…
The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the…
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
In the complex plane, the frequency response of a univariate polynomial is the set of values taken by the polynomial when evaluated along the imaginary axis. This is an algebraic curve partitioning the plane into several connected…
A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion…
Super-stability and strong stability are properties of a matching in the stable matching problem with ties. In this paper, we introduce a common generalization of super-stability and strong stability, which we call non-uniform stability.…
We prove the stability under lamination of a set of real, symmetric 3$\times$3 matrices that can be viewed as a subset of the effective conductivities of a polycrystal. Constructed in a companion paper, such set in combination with several…