Skew-symmetric Research Articles

THE MOORE-PENROSE INVERSE OF TRIDIAGONAL SKEW-SYMMETRIC MATRICES. II

This article is the second part of the work started in the previous publication by the authors [1]. The results presented here relate to deriving closed form expressions for the elements of the Moore-Penrose inverse of tridiagonal real skew-symmetric matrices of odd order. On the base of the formulas obtained, an algorithm that is optimal in terms of the amount of computational efforts is constructed.

Smith Normal Form and the generalized spectral characterization of oriented graphs

Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph Gσ is obtained from a simple undirected graph G by assigning to every edge of G a direction so that Gσ becomes a directed graph. The skew-adjacency matrix of an oriented graph Gσ is a real skew-symmetric matrix S(Gσ)=(sij), where sij=−sji=1 if (i,j) is an arc; sij=sji=0 otherwise. Let Gσ and Hτ be two oriented graphs whose skew-adjacency matrices are S(Gσ) and S(Hτ), respectively. We say Gσ is R-cospectral to Hτ if tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum for any t∈R, where J is the all-ones matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to Gσ is isomorphic to Gσ. Let W(Gσ)=[e,S(Gσ)e,S2(Gσ)e,…,Sn−1(Gσ)e] be the skew-walk-matrix of Gσ, where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if Gσ is a self-converse oriented graph and 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of Gσ we obtain our main result: Let q be a prime and Gσ be a self-converse oriented graph on n vertices with det⁡W(Gσ)≠0. Assume that rankq(W(Gσ))=n−1 if q is an odd prime, and rankq(W(Gσ))=⌈n2⌉ if q=2. If Q is a regular rational orthogonal matrix satisfying QTS(Gσ)Q=S(Hτ) for some oriented graph Hτ, then the level of Q divides dn(W(Gσ))q, where dn(W(Gσ)) is the n-th invariant factor of W(Gσ). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above.

Time-reversal symmetries and equilibriumlike Langevin equations.

Graham has shown [Z. Phys. B 26, 397 (1977)0340-224X10.1007/BF01570750] that a fluctuation-dissipation relation can be imposed on a class of nonequilibrium Markovian Langevin equationsthat admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equationis associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may lose its time-reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time-reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. We find instances where fluctuations due to noise are solely responsible for the dissipation. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.

Affine subspaces of antisymmetric matrices with constant rank

For every n ∈ N and every field K, let A ( n , K ) be the vector space of the antisymmetric ( n × n ) -matrices over K. We say that an affine subspace S of A ( n , K ) has constant rank r if every matrix of S has rank r. Define A antisym K ( n ; r ) = { S | S affine subspace of A ( n , K ) of constant rank r } , a antisym K ( n ; r ) = max { dim ⁡ S ∣ S ∈ A antisym K ( n ; r ) } . In this paper, we prove the following formulas: for n ≥ 2 r + 2 a antisym R ( n ; 2 r ) = ( n − r − 1 ) r ; for n = 2r a antisym R ( n ; 2 r ) = r ( r − 1 ) ; for n = 2r + 1 a antisym R ( n ; 2 r ) = r ( r + 1 ) .

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Research on IMU Calibration Model Based on Polar Decomposition.

As an important deterministic error of the inertial measurement unit (IMU), the installation error has a serious impact on the navigation accuracy of the strapdown inertial navigation system (SINS). The impact becomes more severe in a highly dynamic application environment. This paper proposes a new IMU calibration model based on polar decomposition. Using the new model, the installation error is decomposed into a nonorthogonal error and a misalignment error. The compensation of the IMU calibration model is decomposed into two steps. First, the nonorthogonal error is compensated, and then the misalignment error is compensated. Based on the proposed IMU calibration model, we used a three-axis turntable to calibrate three sets of strapdown inertial navigation systems (SINS). The experimental results show that the misalignment errors are larger than the nonorthogonal errors. Based on the experimental results, this paper proposes a new method to simplify the installation error. This simplified method defines the installation error matrix as an antisymmetric matrix composed of three misalignment errors. The navigation errors caused by the proposed simplified calibration model are compared with the navigation errors caused by the traditional simplified calibration model. The 48-h navigation experiment results show that the proposed simplified calibration model is superior to the traditional simplified calibration model in attitude accuracy, velocity accuracy, and position accuracy.

Anti-diagonalization theory and algorithm of matrices—from skew-symmetric matrices to arbitrary matrices

In this paper, a novel algorithm for anti-diagonalization of skew symmetric matrices via using orthogonal similarity transformations has been introduced. The theory and algorithm about the anti-triangular factorization of skew-symmetric matrices are proved. In the case of skew-symmetric matrices, we prove that the anti-diagonal form is always obtained, resulting in developing a new factorization scheme. Moreover, a theoretical algorithm is given based on the theory of double eigenvector system, which provides all the information for the factorization of arbitrary matrices. Finally, the proposed algorithm is verified effective and efficient through the numerical experiments of anti-diagonalization of matrices over a general number field.

Hessian matrix non-decomposition theorem and its application to nonlinear filtering

In 1983, Roger Brockett gave his invited lecture at the International Congress of Mathematics about the classification of finite dimensional estimation algebra (FDEA), which plays a vitally important role in the classification of finite dimensional filters (FDFs). The Hessian matrix non-decomposition (HMND) theorem was one of the key steps for the complete classification of FDEAs with maximal rank of the arbitrary state space dimension (Yau, 2003; Yau and Hu, 2005). In (Yau et al., 1999), the HMND theorem for homogeneous polynomials of degree 4 was proven and at the same time, it is firstly proposed that there exists a HMND conjecture for the general polynomials of degree 2d with d≥2. In this paper, we consider the Hessian matrix of a nontrivial real homogeneous polynomial at degree 6. For the state space dimension n≤4, we prove such matrix cannot be decomposed in the form Q(x)Q(x)T, where Q(x) is a skew-symmetric matrix whose all elements are homogeneous polynomials of degree 2, and Q(x)T is the transpose of Q(x). The fundamental strategy we use in this paper is that we generalized the techniques proposed by Yau et al. (1999) for the case of degree 4 and derived identities of coefficient matrices for the case of degree 6. For the arbitrary state space dimension n, the HMND theorem is demonstrated under several conditions by certain basic linear algebra techniques. These results can be closely related to the classification of FDEAs with non-maximal rank in the filtering theory.

Vortex Formation Control for Linear Multiagent Systems With Unknown Leader Input and External Disturbances

This brief investigates vortex formation control problem for linear multiagent systems with unknown leader input and external disturbances. This brief proposes a distributed control protocol with antisymmetric matrix which can adjust the interaction between agents to realize the vortex formation of multiagent system. Each agent only communicates with their neighbor agents through the communication topology without requiring the whole system to be fully connected, which makes the control protocol more suitable for large-scale systems. The unknown bounded external disturbance and the input information of leader that may be encountered in practical application are considered in the protocol design. A control term related to the symbolic function is introduced to deal with bounded external disturbances and unknown leader input signal, which can deal with more complex external environment and uncooperative leader in practical applications. Finally, a numerical example is presented to verify the proposed protocol.

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space 𝔤 proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan–Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras).

Coherence invariant maps on order-3 symmetric tensors

In 1940s, Hua established the fundamental theorems of geometry of rectangular matrices, symmetric matrices, skew-symmetric matrices, and hermitian matrices. In 1950s, Jacob generalized Hua's theorems to that of order-2 tensors and symmetric tensors. We extend Jacob's work to maps of order-3 symmetric tensors over C by proving that every surjective coherence invariant map on order-3 symmetric tensors over C is induced by a semilinear isomorphism apart from an additive constant.

Derivative principles for invariant ensembles

In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal or pseudo-diagonal entries as it is the case for Hermitian matrices. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove the uniqueness of the derivative principles.

Almost full rank matrices arising from transitive tournaments

Let F be a field and suppose a := ( a 1 , a 2 , … ) is a sequence of non-zero elements in F . For a tournament T on [ n ] , associate the n × n symmetric matrix M T ( a ) (resp. skew-symmetric matrix M T , skew ( a ) ) with zero diagonal as follows: for i<j, if the edge ij is directed as i → j in T, then set M T ( a ) = a i (resp. M T , skew ( a ) = a i ), else set M T ( a ) = a j (resp. M T , skew ( a ) = a j ). Let M n ( a ) (resp. M n , skew ( a ) ) be the family consisting of all the n × n symmetric matrices M T ( a ) (resp. skew-symmetric matrices M T , skew ( a ) ) as T varies over all tournaments on [ n ] . We show that any matrix in M n ( a ) or M n , skew ( a ) corresponding to a transitive tournament has a rank at least n−1, and this is best possible. This settles in a strong form a conjecture posed in Balachandran et al. [An ensemble of high-rank matrices arising from tournaments; Linear Algebra Appl. 2023;658:310–318]. As a corollary, any matrix in these families has rank at least ⌊ log 2 ⁡ ( n ) ⌋ .

The dual Euler-Rodrigues formula in various mathematical forms and their intrinsic relations

This paper examines the variations and derivations of the dual Euler-Rodrigues formula from various mathematical forms, including the matrix in 6 × 6, the dual matrix, Lie group SE(3) of the exponential map of the Lie algebra se(3), and the dual quaternion conjugation, and investigates their intrinsic connections. Based on the dual Euler-Rodrigues formula, the axis, the dual rotation angle, and the new traces are obtained by using the properties of the skew-symmetric matrices. In decomposing the Chasles’ motion, this paper examines two ways of realization of the motion based on the Mozzi-Chasles’ axis. With the equivalent motion, the paper relates the finite displacement screw matrix, the exponential map, and the dual quaternion conjugation to the dual Euler-Rodrigues formula and reveals their connection with the Mozzi-Chasles axis screw, whose parameters are used to construct the Lie algebra, the dual Euler-Rodrigues formula, and the dual quaternion. Further, using the Mozzi-Chasles axis screw, the paper presents a complete geometrical interpretation, including both the translation and rotation, and associates it with the algebraic presentation. By decomposing the equivalent translation induced by the rotation, the paper presents the mapping between the compound translation and the secondary part of the Mozzi-Chasles axis screw. With this map and the compound translation, the paper hence reveals the intrinsic connection between various presentations of rigid body transformations by formulating them into the dual Euler-Rodrigues formula and presents the relations of the exponential map of the Mozzi-Chasles axis screw to the finite displacement screw matrix and the dual Euler-Rodrigues formula, leading to the understanding of the various forms of a rigid body displacement in correspondence to the dual Euler-Rodrigues formula.

Dual views of the generalized degree ofpurity.

Several approaches and descriptors have been proposed to characterize the purity of coherency or density matrices describing physical states, including the polarimetric purity of 2D and 3D partially polarized waves. This work introduces two interpretations of the degree of purity: one derived from statistics and another from algebra. In the first one, the degree purity is expressed in terms of the mean and standard deviation of the eigenvalue spectrum of the density or coherency matrix of the corresponding state. The second one expresses the purity in terms of two specific measures obtained by decomposing the coherency matrix as a sum of traceless symmetric, antisymmetric, and scalar matrices. We believe these two approaches offer better insights into the purity measure. Furthermore, interesting relations with existing quantities in polarization optics also are described.

Time-convergent random matrices from mean-field pinned interacting eigenvalues

AbstractWe study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system.

Sums of orthogonal, symmetric, and skew-symmetric matrices

An $n$-by-$n$ matrix $A$ is called symmetric, skew-symmetric, and orthogonal if $A^T=A$, $A^T=-A$, and $A^T=A^{-1}$, respectively. We give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type ``"orthogonal $+$ symmetric" in terms of the Jordan form of $A-A^T$. We also give necessary and sufficient conditions on a complex matrix $A$ so that it is a sum of type "orthogonal $+$ skew-symmetric" in terms of the Jordan form of $A+A^T$.

Multiple factorization of skew-symmetric matrices

Multiple factorization of skew-symmetric matrices

Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators

ABSTRACT We present conservativeness criteria for sub-Markovian semigroups generated by divergence type operators with specified infinitesimally invariant measures. The conservativeness criteria in this article are derived by -uniqueness and imply that a given infinitesimally invariant measure becomes an invariant measure. We explore further conditions on the coefficients of the partial differential operators that ensure the uniqueness of the invariant measure beyond the case where the corresponding semigroups are recurrent. A main observation is that for conservativeness and uniqueness of invariant measures in this article, no growth conditions are required for the partial derivatives related to the anti-symmetric matrix of functions that determine a part of the drift coefficient. As stochastic counterparts, our results can be applied to show not only the existence of a pathwise unique and strong solution up to infinity to a corresponding Itô-SDE, but also the existence and uniqueness of invariant measures for the family of strong solutions.

Integrable lattice hierarchies behind Cauchy two-matrix model and Bures ensemble

This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures.