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The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

We study the bilinear forms on the space of measurable $p$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $L^q$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $p$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.

In the paper, we consider the Lotka-Volterra mappings, with a degenerate skew-symmetric matrix, acting in the three-dimensional simplex. We analyze the case, which can be considered as a model for studying airborne diseases. This model includes both recovery and death of the sick. [ABSTRACT FROM AUTHOR]

Many problems in applied mathematics are solved by computing the eigenvalues, singular values, spectral and Euclidean norms of the skew-symmetric matrices. In this article, we first introduce the eigenvalues and singular values of the matrix A = [xi - xj ]in,j=1. Then we obtain the spectral norm and Euclidean norms of A. Finally, some numerical examples are taken to illustrate the correctness of the concluded results. [ABSTRACT FROM AUTHOR]

In many real-world situations, the available data consist of a set of several asymmetric pairwise proximity matrices that collect directed exchanges between pairs of objects measured or observed in a number of occasions (three-way data). To unveil patterns of exchange, a clustering model is proposed that accounts for the systematic differences across occasions. Specifically, the goal is to identify the groups of objects that are primarily origins or destinations of the directed exchanges, and, together, to measure the extent to which these clusters differ across occasions. The model is based on two clustering structures for the objects, which are linked one-to-one and common to all occasions. The first structure assumes a standard partition of the objects to fit the average amounts of the exchanges, while the second one fits the imbalances using an “incomplete” partition of the objects, allowing some to remain unassigned. In addition, to account for the heterogeneity of the occasions, the amounts and directions of exchange between clusters are modeled by occasion-specific weights. An Alternating Least-Squares algorithm is provided. Results from artificial data and a real application on international student mobility show the capability of the model to identify origin and/or destination clusters with common behavior across occasions.

The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.

The paper gives a classification of mixed tournaments corresponding to Lotka-Volterra mappings with a degenerate skew-symmetric matrix acting in S ². For each of the considered cases, the Jacobian spectra at fixed points are studied during classification, and a complete description of the sets P and Q is also given. [ABSTRACT FROM AUTHOR]

The oriented graph associated to a given real skew symmetric matrixSis defined as the weighted oriented graph whose skew adjacency matrix isS. Denote bythe rank of the matrixSand bythe minimum skew rank of the graphG. For any real skew symmetric matrixSwhose associated oriented graphcontains no joining even cycles, we in this paper give a necessary and sufficient condition for, wheremis any integer withanddenotes the matching number of. As an application of our results, we study the minimum skew rank problem, determining the minimum skew rank of graphs containing no joining even cycles, giving a characterization for graphs withand, respectively. [ABSTRACT FROM PUBLISHER]

Skew-symmetric matrices of order 7 defined through the 2-fold vector cross product in ℂ7, and other related matrices, are presented. More concretely, matrix properties, namely invertibility, nullspace, powers and index, are studied. As a consequence, results on vector cross product equations, vector cross product differential equations and vector cross product difference equations in ℂ7 are established.

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. [ABSTRACT FROM AUTHOR]

Purpose of the work is to study the dynamics of the asymptotic behavior of trajectories of discrete Lotka–Volterra dynamical systems with homogeneous tournaments operating in an arbitrary (m − 1)-dimensional simplex. It is known that a dynamic system is an object or a process for which the concept of a state is uniquely defined as a set of certain quantities at a given time, and a law describing the evolution of initial state over time is given. Mainly in questions of population genetics, biology, ecology, epidemiology and economics, systems of nonlinear differential equations describing the evolution of the process under study often arise. Since the Lotka–Volterra equations often arise in life phenomena, the main purpose of the work is to study the trajectories of discrete dynamical Lotka–Volterra systems using elements of graph theory. Methods. In the paper cards of fixed points are constructed for quadratic Lotka–Volterra mappings, that allow describing the dynamics of the systems under consideration. Results. Using cards of fixed points of a discrete dynamical system, criteria for the existence of fixed points with odd nonzero coordinates are given in a particular case, and these results on the location of fixed points of Lotka–Volterra systems are generalized accordingly in the case of an arbitrary simplex. The main results are theorems 5–9, which allow us to describe the dynamics of these systems arising in a number of genetic, epidemiological and ecological models. Conclusion. The results obtained in the paper give a detailed description of the dynamics of the trajectories of Lotka–Volterra maps with homogeneous tournaments. The map of fixed points highlights a specific area in the simplex that is most important and interesting for studying the dynamics of these maps. The results obtained are applicable in environmental problems, for example, to describe and study the cycle of biogens.

The set of all possible determinant values of the sum of a (complex) symmetric matrix and a skew-symmetric matrix with prescribed singular values is determined. This set can also be viewed as the best containment region for the determinant of a square matrix X in terms of the singular values of its symmetric and skew-symmetric parts. The technique is extended to study the principal minors of X. Similar problems for real matrices are considered. [ABSTRACT FROM AUTHOR]

For a fixed configuration in the collinear N-body problem, the existence of the central configurations is determined by a system of linear equations, which in turn is determined by certain Pfaffians for even or odd N in literatures. In this short note, we prove that the Pfaffians of the associate matrices for all even number collinear configurations are nonzero if and only if the extended Pfaffians of the associate matrices for all odd number collinear configurations are nonzero. Therefore, the inverse problem of the collinear central configurations can be answered and each collinear configuration determines a one-parameter family of masses with a fixed total mass if the Pfaffians of the associate matrix for all collinear even number bodies are nonzero. We also make some remarks on the super central configurations and the number of collinear central configurations under different equivalences, especially a lower bound for the number of collinear central configurations under the geometric equivalence.

We study the range [image omitted] 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 ∈ , 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. [ABSTRACT FROM AUTHOR]

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

In [8], the authors obtained the non-zero solutions of the equation A(x)=0,2 1 1, n x E   in Lorentzian space 2 1 1, n E where A is a skew-symmetric matrix corresponding to the linear map A and got normal forms of the skewsymmetric matrix A, depending on the causal characters of the vector x. Taking into consideration the structure of the matrix A, we generate Matlab codes and make some Matlab applications for normal form of skew-symmetric matrix. Also, we give some Matlab codes for the linear first order system of differantial equations which the solution of the system gives rise to integral curves of linear vector fields in such a space. Moreover, we give some application with respect to special case of n and causal characters of the vector x for Matlab.

Sometimes, the same categorical variable is studied over different time periods or across different cohorts at the same time. One may consider, for example, a study of voting behaviour of different age groups across different elections, or the study of the same variable exposed to a child and a parent. For such studies, it is interesting to investigate how similar, or different, the variable is between the two time points or cohorts and so a study of the departure from symmetry of the variable is important. In this paper, we present a method of visualising any departures from symmetry using correspondence analysis. Typically, correspondence analysis uses Pearson’s chi-squared statistic as the foundation for all of its numerical and visual features. In the case of studying the symmetry of a variable, Bowker’s chi-squared statistic, presented in 1948, provides a simple numerical means of assessing symmetry. Therefore, this paper shall discuss how a correspondence analysis can be performed to study the symmetry (or lack thereof) of a categorical variable when Bowker’s statistic is considered. The technique presented here provides an extension to the approach developed by Michael Greenacre in 2000.

We study an optimal boundary control problem (OCP) associated to the linear parabolic equation $y_t - \mathrm{div}(\nabla y + A(x) \nabla y) = f$. The characteristic feature of this equation is the fact that the matrix $A(x) = [a_{ij}(x)]_{i,j=1,...,N}$ is skew-symmetric, $a_{ij}(x) = -a_{ji}(x)$ and belongs to $L^2$-space (rather than $L^{\infty}$). We show that under special choice of matrix $A$ and distribution $f$, a unique solution to the original OCP inherits a singular character of the original matrix $A$ and it can not be attainable by the solutions of the similar OCPs with $L^{\infty}$-approximations of matrix $A$.

The lurch index has recently been introduced in applied kinematics as an integral descriptor of the fluency of the motion of a rigid body in space. It may be defined in different versions, according to the component of motion under investigation. In the present paper, we analyze a rotational lurch index, which describes the fluency of the spin component of motion and whose value depends, through involved relations, on the dynamics of three canonical descriptors of the orientation of a rigid body in space. The aim of the present paper is to offer a closed-form expression of the instantaneous component of the rotational lurch, which leads to the namesake index upon integration and normalization. The closed form of the index is, then, used to evaluate its practical calculation, based on numerical approximations on a number of data sets.

Let an n × n complex matrix A be such that I + A is invertible. The Cayley transform of A , denoted by C (A) , is defined as C (A) = (I + A) − 1 (I − A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]