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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]

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.

This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems. [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. [ABSTRACT FROM AUTHOR]

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]

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.

Given a bilinear form on C n , represented by a matrix A ∈ C n × n , the problem of finding the largest dimension of a subspace of C n such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation X ⊤ A X = B is consistent (here X ⊤ denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation X ⊤ A X = B to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices A ∈ C n × n . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A. [ABSTRACT FROM AUTHOR]

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.

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Investigation of many problems of population genetics, epidemiology, ecology, etc. leads to the study of quadratic simplex mappings, a prominent representative of which are the Lotka–Volterra mappings. In this paper, we consider Lotka–Volterra mappings acting in a four-dimensional simplex with homogeneous tournaments. These mappings are interesting in that they can be offered as a model for studying the elements of living matter coming from the environment, which, having passed through a number of organisms, return to the external environment, and then are again included in the composition of living matter. For a more detailed study of the considered mappings, cards of fixed points are constructed and studied. According to the structure of the card, it is possible to represent the cycles of carbon and phosphorus. [ABSTRACT FROM AUTHOR]

O'Dorney has proved that for any orthogonal matrix U , there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which | s i j | ≤ 1 , where S = (s i j) = Q (U D) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2 − 1 by multiplying by a certain signed permutation matrix, instead of a signature matrix. [ABSTRACT FROM AUTHOR]

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. [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]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, by using gradients of Gateaux differentiable G -increasing functions, we establish some interpolation results for the following G -majorization triangle inequality: (x + y) ↓ ≺ G x ↓ + y ↓ for vectors x , y of an Eaton triple with normal map (⋅) ↓. In particular, we show refinements for the majorization Ky Fan's eigenvalue inequality λ (x + y) ≺ m λ (x) + λ (y) for x , y ∈ H n , where H n is the space of n × n Hermitian matrices and λ (⋅) is the eigenvalue operator on H n. We derive analogous results for an inner product space with its standard norm, and for real skew-symmetric matrices with their singular values. [ABSTRACT FROM AUTHOR]

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.

Minimal realisation problems of odd transfer functions for first-degree (multi-linear) nD single-input single-output discrete systems have been studied, but it has not been well solved. This paper provides a new, different method to solve absolutely minimal realisation problems. By methods of limits and algebraic techniques, without using the symbolic approach by Gröbner basis, the requirements of absolutely minimal realisation are transformed into a system of equations represented by the determinants. Since the equations for first-degree 2D systems are solvable by quadratic equations and the conditions for higher-dimensional realisations can be expressed by the results of 2D systems, the absolutely minimal realisations for nD systems can be found by using the realisations of n(n − 1)/2 2D systems. Furthermore, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived from the Pfaffian function of the skew-symmetric matrix. Finally, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure. [ABSTRACT FROM AUTHOR]

This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question 2 in Yen and Yao (Optimization 60:53-68, 2011) and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation. [ABSTRACT FROM AUTHOR]

A new approach is proposed for deriving the Hessian matrixes of parallel manipulators and their generative mechanisms using the skew-symmetric matrixes of translational/angular velocities and their combination formulae. First, several necessary skew-symmetric matrixes of the translational/ angular velocities and their combination formulae are derived. Second, the differentiations of the sub-Jacobian matrixes are transformed into a multiplication of a general velocity transposition by sub-Hessian matrixes based on the derived formulae. Third, the sub-Hessian matrixes of a typical parallel manipulator and its generative mechanisms (the hybrid hand and the redundant kinematic hybrid manipulator) as well as their kinematic limbs are derived by differentiating the sub-Jacobian matrixes based on the derived the skew-symmetric matrixes of translational/angular velocities and their combination formulae. Finally, the formulae are derived for solving the Hessian matrixes and the accelerations of the typical parallel manipulator and the generative mechanisms. [ABSTRACT FROM AUTHOR]

In this paper we explore the structure of certain generalized isometries of the special orthogonal group SO( n) which are transformations that leave any member of a large class of generalized distance measures invariant. [ABSTRACT FROM AUTHOR]

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 $A$.

We show that the existence of {±1}-matrices having largest possible determinant is equivalent to the existence of certain tournamentmatrices. In particular, we prove a recent conjecture of Armario.We also show that large submatrices of conference matrices are determined by their spectrum. [ABSTRACT FROM AUTHOR]