Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

[EN] Based on the conditions ab(2) = 0 and b pi(ab) is an element of A(d), we derive that (ab)(n), (ba)(n), and ab + ba are all generalized Drazin invertible in a Banach algebra A, where n is an element of N and a and b are elements of A. By using these results, some results on the symmetry representations for the generalized Drazin inverse of ab + ba are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b.This work was supported by the National Natural Science Foundation of China (grant number: 11361009, 61772006,11561015), the Special Fund for Science and Technological Bases and Talents of Guangxi (grant number: 2016AD05050, 2018AD19051), the Special Fund for Bagui Scholars of Guangxi (grant number: 2016A17), the High level innovation teams and distinguished scholars in Guangxi Universities (grant number: GUIJIAOREN201642HAO), the Natural Science Foundation of Guangxi (grant number: 2017GXNSFBA198053, 2018JJD110003), and the open fund of Guangxi Key laboratory of hybrid computation and IC design analysis (grant number: HCIC201607).Qin, Y.; Liu, X.; Benítez López, J. (2019). Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra. Symmetry (Basel). 11(1):1-9. https://doi.org/10.3390/sym11010105S19111González, N. C. (2005). Additive perturbation results for the Drazin inverse. Linear Algebra and its Applications, 397, 279-297. doi:10.1016/j.laa.2004.11.001Zhang, X., & Chen, G. (2006). The computation of Drazin inverse and its application in Markov chains. Applied Mathematics and Computation, 183(1), 292-300. doi:10.1016/j.amc.2006.05.076Castro-González, N., Dopazo, E., & Martínez-Serrano, M. F. (2009). On the Drazin inverse of the sum of two operators and its application to operator matrices. Journal of Mathematical Analysis and Applications, 350(1), 207-215. doi:10.1016/j.jmaa.2008.09.035Qiao, S., Wang, X.-Z., & Wei, Y. (2018). Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra and its Applications, 542, 101-117. doi:10.1016/j.laa.2017.03.014Stanimirovic, P. S., Zivkovic, I. S., & Wei, Y. (2015). Recurrent Neural Network for Computing the Drazin Inverse. IEEE Transactions on Neural Networks and Learning Systems, 26(11), 2830-2843. doi:10.1109/tnnls.2015.2397551Koliha, J. J. (1996). A generalized Drazin inverse. Glasgow Mathematical Journal, 38(3), 367-381. doi:10.1017/s0017089500031803Hartwig, R. E., Wang, G., & Wei, Y. (2001). Some additive results on Drazin inverse. Linear Algebra and its Applications, 322(1-3), 207-217. doi:10.1016/s0024-3795(00)00257-3Djordjević, D. S., & Wei, Y. (2002). Additive results for the generalized Drazin inverse. Journal of the Australian Mathematical Society, 73(1), 115-126. doi:10.1017/s1446788700008508Liu, X., Xu, L., & Yu, Y. (2010). The representations of the Drazin inverse of differences of two matrices. Applied Mathematics and Computation, 216(12), 3652-3661. doi:10.1016/j.amc.2010.05.016Yang, H., & Liu, X. (2011). The Drazin inverse of the sum of two matrices and its applications. Journal of Computational and Applied Mathematics, 235(5), 1412-1417. doi:10.1016/j.cam.2010.08.027Harte, R. (1992). On generalized inverses in C*-algebras. Studia Mathematica, 103(1), 71-77. doi:10.4064/sm-103-1-71-77Djordjevic, D. S., & Stanimirovic, P. S. (2001). On the Generalized Drazin Inverse and Generalized Resolvent. Czechoslovak Mathematical Journal, 51(3), 617-634. doi:10.1023/a:1013792207970Cvetković-Ilić, D. S., Djordjević, D. S., & Wei, Y. (2006). Additive results for the generalized Drazin inverse in a Banach algebra. Linear Algebra and its Applications, 418(1), 53-61. doi:10.1016/j.laa.2006.01.015Liu, X., Qin, X., & Benítez, J. (2016). New additive results for the generalized Drazin inverse in a Banach algebra. Filomat, 30(8), 2289-2294. doi:10.2298/fil1608289lMosić, D., Zou, H., & Chen, J. (2017). The generalized Drazin inverse of the sum in a Banach algebra. Annals of Functional Analysis, 8(1), 90-105. doi:10.1215/20088752-3764461González, N. C., & Koliha, J. J. (2004). New additive results for the g-Drazin inverse. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 134(6), 1085-1097. doi:10.1017/s0308210500003632Mosić, D. (2014). A note on Cline’s formula for the generalized Drazin inverse. Linear and Multilinear Algebra, 63(6), 1106-1110. doi:10.1080/03081087.2014.92296

Forest matrices around the Laplacian matrix

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor

Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

Representations for generalized Drazin inverse of operator matrices over a Banach space

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature

Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open non unitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples.Comment: 29 pages, 4 figures, expanded historical citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht

The generalized inverses of tensors via the C-Product

This paper studies the issues about the generalized inverses of tensors under the C-Product. The aim of this paper is threefold. Firstly, this paper present the definition of the Moore-Penrose inverse, Drazin inverse of tensors under the C-Product. Moreover, the inverse along a tensor is also introduced. Secondly, this paper gives some other expressions of the generalized inverses of tensors by using several decomposition forms of tensors. Finally, the algorithms for the Moore-Penrose inverse, Drazin inverse of tensors and the inverse along a tensor are established

Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt