Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory ...of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases.
Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity, and many ...more. They are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks. Here, we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra. We show here that this matrix function can be derived from physical models that consider the interactions between nearest and next-nearest neighbors in the graph. We first obtain a few mathematical results for the trace of this matrix function when λref=−1 (H−1) for simple graphs as well as for random graphs. We also provide a combinatorial interpretation of this index in terms of subgraphs in the graph, and in terms of the competition pressure among agents in a complex system. Finally, we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992–2002. In both cases we give a clear phenomenological interpretation of the processes described.
In this article we consider the problem of the existence of rational 1,2-pseudo-inverses for rational multivariable matrix-valued functions. We prove that any rational multivariable matrix-valued ...function has rational 1,2-pseudo-inverse and we describe the set of all 1,2-pseudo-inverses of a given function, in terms of rational free parameters. It is shown that the 1,2-pseudo-inverse can have an effective description relative to the Moore–Penrose pseudo-inverse (that in general is not even rational), thus making it an alternative effective solution to the problem of online matrix inversion of large matrices that depend on real-time measured parameters. The rationality of the 1,2-pseudo-inverse is crucial when it should be realized in the physical real world (e.g. in inverse control), in contrast with realizations in computer algorithms (e.g. in image processing and communication systems) where the rationality is not necessary but the effective description of the pseudo-inverse becomes crucial. The results has applications in control systems, robust control, inverse control, mechanical systems, kinematic chains, kinematic networks, multi-degree-of-freedom systems, image processing, signal processing and communication systems.
In this work, a new method to compute the matrix exponential function by using an approximation based on Euler polynomials is proposed. These polynomials are used in combination with the scaling and ...squaring technique, considering an absolute forward-type theoretical error. Its numerical and computational properties have been evaluated and compared with the most current and competitive codes dedicated to the computation of the matrix exponential. Under a heterogeneous test battery and a set of exhaustive experiments, it has been demonstrated that the new method offers performance in terms of accuracy and stability which is as good as or even better than those of the considered methods, with an intermediate computational cost among all of them. All of the above makes this a very competitive alternative that should be considered in the growing list of available numerical methods and implementations dedicated to the approximation of the matrix exponential.
Spectral factorization is a prominent tool with several important applications in various areas of applied science. Wiener and Masani proved the existence of matrix spectral factorization. Their ...theorem has been extended to the multivariable case by Helson and Lowdenslager. Solving the problem numerically is challenging in both situations, and also important due to its practical applications. Therefore, several authors have developed algorithms for factorization. The Janashia-Lagvilava algorithm is a relatively new method for matrix spectral factorization which has proved to be useful in several applications. In this paper, we extend this method to the multivariable case. Consequently, a new numerical algorithm for multivariable matrix spectral factorization is constructed.
Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received ...considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented.
Matrix functions in network analysis Benzi, Michele; Boito, Paola
Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik,
September 2020, 2020-09-00, 20200901, Letnik:
43, Številka:
3
Journal Article
Recenzirano
We review the recent use of functions of matrices in the analysis of graphs and networks, with special focus on centrality and communicability measures and diffusion processes. Both undirected and ...directed networks are considered, as well as dynamic (temporal) networks. Computational issues are also addressed.
In a recent paper (Groenewald et al. 2021 9) we considered an unbounded Toeplitz-like operator TΩ generated by a rational matrix function Ω that has poles on the unit circle T of the complex plane. A ...Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator TΩ, including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of TΩ from this factorization, and hence of the co-kernel, even when TΩ is Fredholm. In the current paper we provide a formula for the dimension of the kernel of TΩ under an additional assumption on the Wiener-Hopf type factorization. In the case that Ω is a 2×2 matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the 2×2 case is partially extended to the case of matrix functions of arbitrary size.
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