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Andrii Dmytryshyn - One of the best experts on this subject based on the ideXlab platform.

  • Generic Skew-Symmetric Matrix polynomials with fixed rank and fixed odd grade
    Linear Algebra and its Applications, 2018
    Co-Authors: Andrii Dmytryshyn, Froilán M. Dopico
    Abstract:

    We show that the set of m×m complex Skew-Symmetric Matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of Matrix polynomials ...

  • Generic Skew-Symmetric Matrix polynomials with fixed rank and fixed odd grade
    arXiv: Rings and Algebras, 2017
    Co-Authors: Andrii Dmytryshyn, Froilán M. Dopico
    Abstract:

    We show that the set of $m \times m$ complex Skew-Symmetric Matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of Matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex Skew-Symmetric Matrix polynomials of odd grade $d$ and rank at most $2r$. In particular, this result includes the case of Skew-Symmetric Matrix pencils ($d=1$).

  • Structure preserving stratification of Skew-Symmetric Matrix polynomials
    Linear Algebra and its Applications, 2017
    Co-Authors: Andrii Dmytryshyn
    Abstract:

    We study how elementary divisors and minimal indices of a Skew-Symmetric Matrix polynomial of odd degree may change under small perturbations of the Matrix coefficients. We investigate these change ...

  • Orbit closure hierarchies of Skew-Symmetric Matrix pencils
    SIAM Journal on Matrix Analysis and Applications, 2014
    Co-Authors: Andrii Dmytryshyn, Bo Kågström
    Abstract:

    We study how small perturbations of a Skew-Symmetric Matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of Skew-Symmetric Matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a Skew-Symmetric Matrix pencil contains the congruence orbit (or bundle) of another Skew-Symmetric Matrix pencil. The developed theory relies on our main theorem stating that a Skew-Symmetric Matrix pencil $A-\lambda B$ can be approximated by pencils strictly equivalent to a Skew-Symmetric Matrix pencil $C-\lambda D$ if and only if $A-\lambda B$ can be approximated by pencils congruent to $C-\lambda D$.

  • ORBIT CLOSURE HIERARCHIES OF Skew-Symmetric
    2014
    Co-Authors: Matrix Pencils, Andrii Dmytryshyn, Bo K
    Abstract:

    We study how small perturbations of a Skew-Symmetric Matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of Skew-Symmetric Matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a Skew-Symmetric Matrix pencil contains the congruence orbit (or bundle) of another Skew-Symmetric Matrix pencil. The developed theory relies on our main theorem stating that a Skew-Symmetric Matrix pencil A − λB can be approximated by pencils strictly equivalent to a Skew-Symmetric Matrix pencil C − λD if and only if A − λB can be approximated by pencils congruent to C − λD.

Bo Kågström - One of the best experts on this subject based on the ideXlab platform.

Yusuf Yayli - One of the best experts on this subject based on the ideXlab platform.

  • Chord Properties of Some Special Curves in Euclidean Space
    2020
    Co-Authors: Emre Öztürk, Yusuf Yayli
    Abstract:

    In this paper, we define some special curves through the chord that combines two different points of the curve on it, and we examine relations of these curves each other. Especially, these curves have been characterized by their unit tangent vector field itself with symmetric and skew symmetric Matrix. Moreover, we show that these curves are the geodesics of the isoparametric surfaces such as spheres, right circular cylinders and spherical cylinders.

  • the geometrical and algebraic interpretations of euler rodrigues formula in minkowski 3 space
    International Journal of Geometric Methods in Modern Physics, 2016
    Co-Authors: Derya Kahveci, Yusuf Yayli
    Abstract:

    The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in ℝ13 and angle, the spatial displacement is represented by a 3 × 3 semi-orthogonal rotation Matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-Skew-Symmetric Matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

  • The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space
    International Journal of Geometric Methods in Modern Physics, 2016
    Co-Authors: Derya Kahveci̇, Yusuf Yayli
    Abstract:

    The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in ℝ13 and angle, the spatial displacement is represented by a 3 × 3 semi-orthogonal rotation Matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-Skew-Symmetric Matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

  • FORMULAS FOR THE EXPONENTIAL OF A SEMI SKEW- SYMMETRIC Matrix OF ORDER 4
    Mathematical & Computational Applications, 2005
    Co-Authors: Levent Kula, Murat Kemal Karacan, Yusuf Yayli
    Abstract:

    In this paper the formula of the exponential Matrix e A when A is a semi Skew-Symmetric real Matrix of order 4 is derived. The formula is a generalization of the Rodrigues formula for Skew-Symmetric matrices of order 3 in Minkowski 3-space.

Tin-yau Tam - One of the best experts on this subject based on the ideXlab platform.

  • A range associated with skew symmetric Matrix
    Linear and Multilinear Algebra, 2011
    Co-Authors: Xuhua Liu, Dawit G. Tadesse, Tin-yau Tam
    Abstract:

    We study the range where A is an n × n complex skew symmetric Matrix. It is a compact convex set. Power inequality s(A2k+1) ≤ s2k+1(A), k ∈ ℕ, for the radius s(A) ≔ maxξ∈S(A) |ξ| is proved. When n = 3, 4, 5, 6, relations between S(A) and the classical numerical range and the k-numerical range are given. Axiomatic characterization of S(A) is given. Sharp points and extreme points are studied.

  • Pfaffian and decomposable numerical range of a complex skew symmetric Matrix
    Linear and Multilinear Algebra, 2009
    Co-Authors: Wai-shun Cheung, Tin-yau Tam
    Abstract:

    In the literature it is known that the decomposable numerical range of is not necessarily convex. But it is not known whether is star-shaped. We construct a symmetric unitary Matrix such that the decomposable numerical range is not star-shaped and hence not simply connected. We then consider a real analog and show that is star-shaped if is skew symmetric. Such a star-shapedness result is also true for the Pfaffian numerical range .

  • Partial Superdiagonal Elements and Singular Values of a Complex Skew-Symmetric Matrix
    SIAM Journal on Matrix Analysis and Applications, 1998
    Co-Authors: Tin-yau Tam
    Abstract:

    Let A be an m x m complex Skew-Symmetric Matrix with singular values $s_1\ge s_1 \ge s_2 \ge s_2\ge \cdots \ge s_n\ge s_n \ge 0$, where n = [m/2]$. We consider the sets ${\D}_p(A) = {diag (UTAU[1,..., p|n+1, ..., n+p]): U\in U(m)\}$, $p = 1, \dots , n$, where $U(m)$ denotes the unitary group. We prove that when m=2n and p=n, d = (d1,..., dn) \in {\D}_n(A)$ if and only if \begin{eqnarray*} \sum_{i=1}^k |d_i|&\le &\sum_{i=1}^k s_i, \qquad k = 1, \dots , n,\\ \sum_{i=1}^{n-1} |d_i| - |d_n| &\le &\sum_{i=1}^{n-1} s_i - s_n, \end{eqnarray*} after rearranging the entries of d in descending order with respect to absolute value. The set is not convex in general. The inequalities are identical to those of Thompson--Sing's theorem on the diagonal elements and the singular values of an n x n complex Matrix. All other cases, i.e., (1) $m=2n+1$ and $1\le p\le n$ and (2) $m=2n$ and $1\le p

Leonid Gurvits - One of the best experts on this subject based on the ideXlab platform.

  • Control of nonholonomic systems and decomposition of skew symmetric matrices
    Proceedings of 32nd IEEE Conference on Decision and Control, 1
    Co-Authors: Leonid Gurvits
    Abstract:

    We consider in this paper some applications of two problems to nonlinear control and linear algebra. Problem A concerns a skew symmetric Matrix with elements belonging to a commutative algebra. It is desirable to minimize the Matrix. Problem B concerns optimal decomposition for a skew symmetric Matrix with complex elements and positive numbers. It is shown how one may obtain new spectral inequalities for matrices using an optimal control problem. The crucial point in this reduction is a time scalability property. We would like to point out here that as the development of linear control theory gave a push to the development of linear algebra, nonholonomic control theory will enrich polylinear algebra. >