scholarly journals The Drazin inverse matrix modification formulae with Peirce corners

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2605-2616
Author(s):  
Daochang Zhang ◽  
Dijana Mosic ◽  
Jianping Hu
Keyword(s):  
Inverse Matrix ◽  
Drazin Inverse ◽  
Special Cases ◽  
Matrix Modification ◽  
Additive Perturbation ◽  
Special Matrix

Our motivation is to derive the Drazin inverse matrix modification formulae utilizing the Drazin inverses of adequate Peirce corners under some special cases, and the Drazin inverse of a special matrix with an additive perturbation. As applications, several new results for the expressions of the Drazin inverses of modified matrices A ?? CB and A ?? CDdB are obtained, and some well known results in the literature, as the Sherman-Morrison-Woodbury formula and Jacobson?s Lemma, are generalized.

scholarly journals Additive results for the generalized Drazin inverse

2002 ◽  
Vol 73 (1) ◽  
pp. 115-126 ◽  
Author(s):  
Dragan S. Djordjević ◽  
Yimin Wei
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Drazin Inverse ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Special Cases ◽  
Arbitrary Complex ◽  
Additive Perturbation ◽  
Generalized Drazin Inverse ◽  
Banach Space Operators

AbstractAdditive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Fault Current Constraint Transmission Expansion Planning Based on the Inverse Matrix Modification Lemma and a Valid Inequality

Energies ◽  
2019 ◽  
Vol 12 (24) ◽  
pp. 4769
Author(s):  
Sungwoo Lee ◽  
Hyoungtae Kim ◽  
Tae Hyun Kim ◽  
Hansol Shin ◽  
Wook Kim
Keyword(s):  
Current Level ◽  
Inverse Matrix ◽  
Valid Inequality ◽  
Fault Current ◽  
Impedance Matrix ◽  
Expansion Planning ◽  
Transmission Expansion Planning ◽  
Fault Currents ◽  
Transmission Expansion ◽  
Matrix Modification

In the transmission expansion planning (TEP) problem, it is challenging to consider a fault current level constraint due to the time-consuming update process of the bus impedance matrix, which is required to calculate the fault currents during the search for the optimal solution. In the existing studies, either a nonlinear update equation or its linearized version is used to calculate the updated bus impedance matrix. In the former case, there is a problem in that the mathematical formulation is derived in the form of mixed-integer nonlinear programming. In the latter case, there is a problem in that an error due to the linearization may exist and the change of fault currents in other buses that are not connected to the new transmission lines cannot be detected. In this paper, we use a method to obtain the exact updated bus impedance matrix directly from the inversion of the bus admittance matrix. We propose a novel method based on the inverse matrix modification lemma (IMML) and a valid inequality is proposed to find a better solution to the TEP problem with fault current level constraint. The proposed method is applied to the IEEE two-area reliability test system with 96 buses to verify the performance and effectiveness of the proposed method and we compare the results with the existing methods. Simulation results show that the existing TEP method based on impedance matrix modification method violates the fault current level constraint in some buses, while the proposed method satisfies the constraint in all buses in a reasonable computation time.

scholarly journals On the Approximation of Functions From Lp by Some Special Matrix Means of Fourier Series

2015 ◽  
Vol 55 (1) ◽  
pp. 81-90
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix ◽  
Norm Approximation ◽  
Appl Math

Abstract The results corresponding to some theorems of S. Lal [Appl. Math. and Comput. 209 (2009), 346-350] and the results of W. Łenski and B. Szal [Banach Center Publ., 95, (2011), 339-351] are shown. The better degrees of pointwise approximation than these in mentioned papers by another assumptions on summability methods for considered functions are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated.

scholarly journals Cayley-Hamilton theorem for Drazin inverse matrix and standard inverse matrices

2016 ◽  
Vol 64 (4) ◽  
pp. 793-797
Author(s):  
T. Kaczorek
Keyword(s):  
Characteristic Polynomial ◽  
Inverse Matrix ◽  
Drazin Inverse ◽  
Nonsingular Matrix ◽  
Singular Matrix ◽  
Hamilton Equations

Abstract The classical Cayley-Hamilton theorem is extended to Drazin inverse matrices and to standard inverse matrices. It is shown that knowing the characteristic polynomial of the singular matrix or nonsingular matrix, it is possible to write the analog Cayley-Hamilton equations for Drazin inverse matrix and for standard inverse matrices.

scholarly journals A note on computing the generalized inverseA T,S (2)of a matrixA

2002 ◽  
Vol 31 (8) ◽  
pp. 497-507 ◽  
Author(s):  
Xiezhang Li ◽  
Yimin Wei
Keyword(s):  
Iterative Methods ◽  
Half Plane ◽  
Generalized Inverse ◽  
Null Space ◽  
Drazin Inverse ◽  
Numerical Examples ◽  
Special Cases ◽  
Computational Procedures

The generalized inverseA T,S (2)of a matrixAis a{2}-inverse ofAwith the prescribed rangeTand null spaceS. A representation for the generalized inverseA T,S (2)has been recently developed with the conditionσ (GA| T)⊂(0,∞), whereGis a matrix withR(G)=TandN(G)=S. In this note, we remove the above condition. Three types of iterative methods forA T,S (2)are presented ifσ(GA|T)is a subset of the open right half-plane and they are extensions of existing computational procedures ofA T,S (2), including special cases such as the weighted Moore-Penrose inverseA M,N †and the Drazin inverseAD. Numerical examples are given to illustrate our results.

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