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Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition (SVD) of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-k approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore--Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function. [ABSTRACT FROM AUTHOR]

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form x + jy with j² = 1, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a 2 × 2 split-complex matrix in three cases: i. Δ = 0, ii. Δ ̸= 0 and Δ is not null split-complex number, iii. Δ ̸= 0 and Δ is a null split-complex number where Δ = (trA)2 - 4 detA. [ABSTRACT FROM AUTHOR]

The development of algorithmic differentiation (AD) tools focuses mostly on handling floating point types in the target language. Taping optimizations in these tools mostly focus on specific operations like matrix vector products. Aggregated types like std::complex are usually handled by specifying the AD type as a template argument. This approach provides exact results, but prevents the use of expression templates. If AD tools are extended and specialized such that aggregated types can be added to the expression framework, then this will result in reduced memory utilization and improve the timing for applications where aggregated types such as complex number or matrix vector operations are used. Such an integration requires a reformulation of the stored data per expression and a rework of the tape evaluation process. We will demonstrate the overheads on a synthetic benchmark and show the improvement when aggregated types are handled properly by the expression framework of the AD tool. [ABSTRACT FROM AUTHOR]

Let I n be the n × n identity matrix and J = 0 I n − I n 0 . A matrix A is called symplectic if A T J A = J. A symplectic matrix A is a commutator of symplectic involutions if A = X Y X − 1 Y − 1 , where X and Y are symplectic matrices satisfying X 2 = Y 2 = I. In this article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions. [ABSTRACT FROM AUTHOR]

This paper proposes a complex number representation method for dynamically recording robot positions and develops an optimization strategy for measuring and minimizing inter-robot distances in real-time. Based on this, the optimization strategy is transformed into finding the theoretical solution of the complex time-varying matrix equation by combining the leader–follower frameworks, and a complex zeroing neural dynamics (CZND) model is proposed for this objective. Meanwhile, the effectiveness of the proposed CZND model and the ideal trajectories of the follower robots are ensured and given through rigorous theoretical analyses. Subsequently, these theoretical findings are validated by two numerical experiments (four cases). In addition, the superiority of the CZND model for inter-robot management tasks is comparatively analyzed from three perspectives: initial positions, number of follower robots, and design parameters. Compared to conventional gradient-based neural network (CGNN) methods, the proposed CZND model, based on a dynamic error function (which is not required to be non-negative as in CGNN), demonstrates improved handling of the dynamic behavior of the system by utilizing the first-order time derivatives of matrices, resulting in lower residual errors in the implementation of inter-robot management. Specifically, for the same inter-robot management task, the steady-state error implemented by CZND is on the order of 1 0 − 4 , whereas for CGNN, it is on the order of 1 0 − 1 , indicating a significantly larger lagging error. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

• Using the structural dynamic modification theory, the general damped system is considered a perturbation of the un-damped system, where the modification is defined by the damping matrix. C • The complexity of the mode shapes depends on the level of orthogonality of the normal mode shapes with respect to damping matrix. C. • The modal parameters of general damped systems (poles, complex modal masses, length of mode shapes) can be expressed as linear combination of the corresponding un-damped modal parameters. • If the mass density of the system is constant, the modal mass is always equal to the product between the total mass of the structure and the magnitude of the length squared, but with a phase to introduce a complex number. In this paper the modal mass is considered in the case of general damping where it is well-known that the modal mass becomes a complex valued quantity. The aim of this paper is to present a definition of the complex modal mass, which leads to an easier physical understanding of this modal parameter. It is demonstrated that based on the structural modification theory, the complex modal mass can be expressed as a linear combination of the un-damped modal masses, where a complex coefficient matrix can be defined that depends on the general damping matrix and on the mode shapes of the damped and the un-damped system. An important finding is that an apparent mass can be defined so that the modal mass is always equal to the product between the apparent mass and the length squared. The paper includes a 2-DOF example that illustrates the influence of the amount of general damping on modal quantities that are normally considered non-sensitive to damping. [ABSTRACT FROM AUTHOR]

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to R I I -recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices. [ABSTRACT FROM AUTHOR]

A complex unit gain graph (T -gain graph) Φ = (G , φ) is a graph where the function φ assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. A T -gain graph is balanced if the product of the edge gains of each oriented cycle (if any) is 1. We propose two notions of gain distance matrices D < max (Φ) and D < min (Φ) of a T -gain graph Φ, for any ordering '<' of the vertex set. We characterize the gain graphs for which the gain distance matrices are independent of the vertex ordering. We show D < max (Φ) = D < min (Φ) holds for the standard ordering of the vertices if and only if the same holds for any ordering of the vertices, and we call such T -gain graphs as distance compatible gain graphs. We characterize the distance compatible gain graphs whose gain distance matrices are cospectral with the distance matrix of the underlying graph. Besides, we introduce the notion of positively weighted T -gain graphs and establish an equivalent condition for the balance of a T -gain graph. Acharya's and Stanić's spectral criteria for balance are deduced as a consequence. Besides, we obtain some spectral characterizations for the balance of a T -gain graph in terms of the gain distance matrices. Finally, we characterize the distance compatible bipartite T -gain graphs. We show a T -gain graph Φ is distance compatible if and only if every block of Φ is distance compatible. [ABSTRACT FROM AUTHOR]

In this paper we extend Theorems 3.1 and 3.2 of Xu et al. (2020) to more general cases and propose analytic solutions of the following constrained matrix maximization problems with unitary constraints: max S k S k H = I n , L k L k H = I m det c I m + ∏ k = 1 s A k S k B k L k and max S k S k H = I n , L k L k H = I m tr c I m + ∏ k = 1 s A k S k B k L k , where c is a complex number, I m denotes the m -order identity matrix, A k , B k are m × n and n × m complex matrices, and det (⋅) , tr (⋅) denote the matrix determinant and trace functions. This is a non-convex nonlinear constrained matrix maximization problem. The new results improve the corresponding existing ones in Xu et al. (2020). [ABSTRACT FROM AUTHOR]