Skew-symmetric Research Articles

Homothetic Cayley Formula and its Applications

In this study the homothetic Cayley Formula has been defined for an antisymmetric matrix and the relation between an antisymmetric matrix and a vector has been given. Hence Rodrigues and Euler parameters have been obtained for homothetic motions. On the other hand, some results and applications have been achieved by using homothetic rotation matrix.

On the Hilbert scheme of degeneracy loci of twisted differential forms

We prove that, for 3 < m < n-1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Omega(2), the twisted cotangent bundle on P^{n-1}. For 3=m<n-1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parametrized by the Pfaffians of linear skew-symmetric matrices of order n.

An Elementary Derivation of the Matrix Elements of Real Irreducible Representations of so(3)

Using elementary techniques, an algorithmic procedure to construct skew-symmetric matrices realizing the real irreducible representations of so(3) is developed. We further give a simple criterion that enables one to deduce the decomposition of an arbitrary real representation R of so(3) into real irreducible components from the characteristic polynomial of an arbitrary representation matrix.

Boundary integral formulation for interfacial cracks in thermodiffusive bimaterials

An original boundary integral formulation is proposed for the problem of a semi-infinite crack at the interface between two dissimilar elastic materials in the presence of heat flows and mass diffusion. Symmetric and skew-symmetric weight function matrices are used together with a generalized Betti's reciprocity theorem in order to derive a system of integral equations that relate the applied loading, the temperature and mass concentration fields, the heat and mass fluxes on the fracture surfaces and the resulting crack opening. The obtained integral identities can have many relevant applications, such as for the modelling of crack and damage processes at the interface between different components in electrochemical energy devices characterized by multi-layered structures (solid oxide fuel cells and lithium ions batteries).

The enhanced principal rank characteristic sequence for skew-symmetric matrices

The enhanced principal rank characteristic sequence (epr-sequence) was originally defined for an n×n real symmetric matrix or an n×n Hermitian matrix. Such a sequence is defined to be ℓ1ℓ2⋯ℓn where ℓk is A, S, or N depending on whether all, some, or none of the matrix principal minors of order k are nonzero. Here we give a complete characterization of the attainable epr-sequences for real skew-symmetric matrices. With the constraint that ℓk=0 if k is odd, we show that nearly all epr-sequences are attainable by skew-symmetric matrices, which is in contrast to the case of real symmetric or Hermitian matrices for which many epr-sequences are forbidden.

Arithmetic differential equations on $$GL_n$$ G L n , II: arithmetic Lie–Cartan theory

Motivated by the search of a concept of linearity in the theory of arithmetic differential equations (Buium in Arithmetic differential equations. Math. surveys and monographs, vol 118. American Mathematical Society, Providence, 2005), we introduce here an arithmetic analogue of Lie algebras, of Chern connections, and of Maurer–Cartan connections. Our arithmetic analogues of Chern connections are certain remarkable lifts of Frobenius on the p-adic completion of \(GL_n\) which are uniquely determined by certain compatibilities with the “outer” involutions defined by symmetric (respectively, antisymmetric) matrices. The Christoffel symbols of our arithmetic Chern connections will involve a matrix analogue of the Legendre symbol. The analogues of Maurer–Cartan connections can then be viewed as families of “linear” flows attached to each of our Chern connections. We will also investigate the compatibility of lifts of Frobenius with the inner automorphisms of \(GL_n\); in particular, we will prove the existence and uniqueness of certain arithmetic analogues of “isospectral flows” on the space of matrices.

Skew-symmetric matrices and their principal minors

In this article, we study the following problem. Given a positive integer k, what is the relationship between two matrices with entries in a field K and having equal corresponding principal minors of order at most k? Our main theorem improves a result of Loewy [13] for skew-symmetric matrices all of whose off-diagonal entries are nonzero.

A normalizing isospectral flow on complex Hessenberg matrices

We study an isospectral flow (Lax flow) that provides an explicit deformation from upper Hessenberg complex matrices to normal matrices, extending to the complex case and to the case of normal matrices the results of [2]. The Lax flow is given bydAdt=[[A†,A]du,A], where brackets indicate the usual matrix commutator, [A,B]:=AB−BA, A† is the conjugate transpose of A and the matrix [A†,A]du is the matrix equal to [A†,A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A0 is an upper Hessenberg matrix with simple spectrum, then limt→+∞⁡A(t) exists and it is a normal upper Hessenberg matrix isospectral to A0 and if the spectrum of A0 is contained in a line in the complex plane, then the ω-limit set is actually a tridiagonal normal matrix. Furthermore, we show that this flow is also the solution of an infinite time horizon optimal control problem and we prove that it can be used to construct even dimensional real skew-symmetric tridiagonal matrices with given simple spectrum, and with given signs pattern for the codiagonal elements.

Strongly primitive species with potentials I: mutations

Motivated by the mutation theory of quivers with potentials developed by Derksen–Weyman–Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.

A comparison of different solutions of the Bursa–Wolf model and of the 3D, 7-parameter datum transformation

The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial coordinate systems utilizing common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry the most often used procedure to move from one coordinate system to the other is the 3D, 7 parameter (Helmert) transformation. Up to recent times this task was solved either by iteration, or by applying the Bursa–Wolf model. Producers of GPS/GNSS receivers install these algorithms in their systems to achieve a quick processing of data. But nowadays algebraic methods of mathematics give closed form solutions of this problem, which require high level computer technology background. In everyday usage, the closed form solutions are much more simple and have a higher precision than earlier procedures and thus it can be predicted that these new solutions will find their place in the practice. The paper discusses various methods for calculating the scale factor and it also compares solutions based on quaternion with those that are based on rotation matrix making use of skew-symmetric matrix.

Hamiltonian structure, Darboux transformation for a soliton hierarchy associated with Lie algebra so(4,ℂ )**Project supported by the National Natural Science Foundation of China (Grant Nos. 61170183 and 11271007), SDUST Research Fund, China (Grant No. 2014TDJH102), the Fund from the Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province, the Promotive Research Fund for

In this paper, we first introduce a Lie algebra of the special orthogonal group, g = so(4, ), whose elements are 4 × 4 trace-free, skew-symmetric complex matrices. As its application, we obtain a new soliton hierarchy which is reduced to AKNS hierarchy and present its bi-Hamiltonian structure and Liouville integrability. Furthermore, for one of the equations in the resulting hierarchy, we construct a Darboux matrix T depending on the spectral parameter λ.

Lower bound for ranks of invariant forms

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad–Schreyer–Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.

Tucker's theorem for almost skew-symmetric matrices and a proof of Farkas' lemma

A real square matrix A is said to be almost skew-symmetric if its symmetric part has rank one. In this article certain fundamental questions on almost skew-symmetric matrices are considered. Among other things, necessary and sufficient conditions on the entries of a matrix in order for it to be almost skew-symmetric are presented. Sums and subdirect sums are studied. Certain new results for the Moore–Penrose inverse of an almost skew-symmetric matrix are proved. An interesting analogue of Tucker's theorem for skew-symmetric matrices is derived for almost skew-symmetric matrices. Surprisingly, this analogue leads to a proof of Farkas' lemma.

A quadratic Poisson Gel’fand-Kirillov problem in prime characteristic

The quadratic Poisson Gel’fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polynomial algebra K [ X 1 , … , X n ] \mathbb {K}[X_1, \dots , X_n] with Poisson bracket defined by { X i , X j } = λ i j X i X j \{X_i,X_j\}=\lambda _{ij} X_iX_j for some skew-symmetric matrix ( λ i j ) ∈ M n ( K ) (\lambda _{ij}) \in M_n(\mathbb {K}) . This problem was studied in 2011 by Goodearl and Launois over a field of characteristic 0 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel’fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel’fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings. We introduce the concept of higher Poisson derivation, which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic. When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel’fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel’fand-Kirillov problem.

Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems

In this paper we study differential operators of the form \begin{align*} \left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2, \end{align*} for matrices $A,B\in\mathbb{C}^{N,N}$, where the eigenvalues of $A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx, \nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators such as $\mathcal{L}_{\infty}$ arise as linearizations at rotating waves in time-dependent reaction diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that $A$ and $B$ can be diagonalized simultaneously we construct a heat kernel matrix $H(x,\xi,t)$ of $\mathcal{L}_{\infty}$ that solves the evolution equation $v_t=\mathcal{L}_{\infty}v$. In the following we study the Ornstein-Uhlenbeck semigroup \begin{align*} \left[ T(t)v\right](x) = \int_{\mathbb{R}^d} H(x,\xi,t) v(\xi) d\xi,\,x \in \mathbb{R}^d,\, t>0, \end{align*} in exponentially weighted function spaces. This is used to derive resolvent estimates for $\mathcal{L}_{\infty}$ in exponentially weighted $L^p$-spaces $L^p_{\theta} (\mathbb{R}^d,\mathbb{C}^N)$, $1\leqslant p<\infty$, as well as in exponentially weighted $C_{\mathrm{b}}$-spaces $C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$.

On graphs of minimum skew rank 4

The zero–nonzero pattern of a skew-symmetric matrix defines a graph. The minimum rank of all real skew-symmetric matrices with a given graph is studied. The probabilistic method is used to show that for sufficiently large , there is a regular graph of order whose complement has girth at least 5 and for which every skew-symmetric matrix with graph has rank at least 6. This settles an open problem. It is also proved that almost all graphs on vertices do not allow a skew-symmetric matrix whose rank is less than .

Numerical Stability in the Presence of Variable Coefficients

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 differentiation matrices for univariate finite-difference methods. We prove that, in order to sustain a skew-symmetric differentiation matrix of order \(p\ge 2\), a grid must satisfy \(2p-3\) polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth \(2p-1\), which can be derived in a constructive manner. Some applications require not just skew-symmetry, but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further. Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods. We conclude the paper with a list of open problems and their discussion.

CLUSKEXT: CLUstering model for SKew-symmetric data including EXTernal information

A CLUstering model for SKew-symmetric data including EXTernal information (CLUSKEXT) is proposed, which relies on the decomposition of a skew-symmetric matrix into within and between cluster effects which are further decomposed into regression and residual effects when possible external information on the objects is available. In order to fit the imbalances between objects, the model jointly searches for a partition of objects and appropriate weights which are in turn linearly linked to the external variables. The proposal is fitted in a least-squares framework and a decomposition of the fit is derived. An appropriate Alternating Least-Squares algorithm is provided to fit the model to illustrative real and artificial data.

Understanding rigid body motion in arbitrary dimensions

Why would anyone wish to generalize the already unappetizing subject of rigid body motion to an arbitrary number of dimensions? At first sight, the subject seems to be both repellent and superfluous. The author will try to argue that an approach involving no specific three-dimensional constructs is actually easier to grasp than the traditional approach and might thus be generally useful to understand rigid body motion both in three dimensions and in the general case. Specific differences between the viewpoint suggested here and the usual one include the following: here angular velocities are systematically treated as antisymmetric matrices, a symmetric tensor I quite different from the moment of inertia tensor plays a central role, whereas the latter is shown to be a far more complex object, namely a tensor of rank four. A straightforward way to define it is given. The Euler equation is derived and the use of Noether’s theorem to obtain conserved quantities is illustrated. Finally the equations of motion for a heavy top as well as for two bodies linked by a spherical joint are derived to display the simplicity and the power of the method.

New rigorous perturbation bounds for the Cholesky-like factorization of skew-symmetric matrix

Some new rigorous perturbation bounds for the Cholesky-like factorization of skew-symmetric matrix are obtained when the original matrix has the normwise or componentwise perturbations, where the componentwise perturbation has the form of backward rounding error for the Cholesky-like factorization algorithm. These bounds can be much tighter than the existing one. Two numerical examples are provided to illustrate these results.