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Journal articles on the topic 'Generalized Matrix Inverse (or g-inverse)'

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Published: 5 June 2025

Last updated: 15 July 2025

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1

Udwadia, Firdaus E. "When Does a Dual Matrix Have a Dual Generalized Inverse?" Symmetry 13, no. 8 (2021): 1386. http://dx.doi.org/10.3390/sym13081386.

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This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

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2

Zhou, Mengmeng, Jianlong Chen, and Néstor Thome. "The W-weighted Drazin-star matrix and its dual." Electronic Journal of Linear Algebra 37, no. 37 (2021): 72–87. http://dx.doi.org/10.13001/ela.2021.5389.

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After decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems.

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3

Cao, Xiaofei, Yuyue Huang, Xue Hua, Tingyu Zhao, and Sanzhang Xu. "Matrix inverses along the core parts of three matrix decompositions." AIMS Mathematics 8, no. 12 (2023): 30194–208. http://dx.doi.org/10.3934/math.20231543.

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<abstract><p>New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $ be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively, where EP denotes the EP matrix. A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $. One can prove that, the Drazin inverse is the inverse along $ A_{1} $, the weak group inverse is the inverse along $ \hat{A}_{1} $ and the core-EP inverse is the inverse along $ \tilde{A}_{1} $. A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively. In addition, we proved that the Drazin inverse of $ A $ is the inverse of $ A $ along $ U $ and $ A_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the weak group inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \hat{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the core-EP inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \tilde{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $. Let $ X_{1} $, $ X_{4} $ and $ X_{7} $ be the generalized inverses along $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $, respectively. In the last section, some useful examples were given, which showed that the generalized inverses $ X_{1} $, $ X_{4} $ and $ X_{7} $ were different generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.</p></abstract>

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4

Chen, Yang, Kezheng Zuo, and Zhimei Fu. "New characterizations of the generalized Moore-Penrose inverse of matrices." AIMS Mathematics 7, no. 3 (2022): 4359–75. http://dx.doi.org/10.3934/math.2022242.

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<abstract><p>Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.</p></abstract>

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5

Liu, Xifu, and Rouyue Fang. "Notes on Re-nnd generalized inverses." Filomat 29, no. 5 (2015): 1121–25. http://dx.doi.org/10.2298/fil1505121l.

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Motivated by a recent paper, in which the authors studied Re-nnd {1,3}-inverse, {1,4}-inverse and {1,3,4}-inverse of a square matrix, in this paper, we establish some equivalent conditions for the existence of Re-nnd {1,2,3}-inverse, {1,2,4}-inverse and {1,3,4}-inverse. Furthermore, some expressions of these generalized inverses are presented.

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6

Xu, Sanzhang, and Dingguo Wang. "New characterizations of the generalized B-T inverse." Filomat 36, no. 3 (2022): 945–50. http://dx.doi.org/10.2298/fil2203945x.

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Characterizations and explicit expressions of the generalized B-T inverse are given, this generalized inverse exists for any square matrix and any integer. The relationships between the generalized B-T inverse and some well-known generalized inverses are investigated. Moreover, an explicit formula of the generalized B-T inverse is given by using Hartwig-Spindelb?ck decomposition.

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7

Liu, Xiaoji, Hongwei Jin, and Jelena Višnjić. "Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms." Mathematical Problems in Engineering 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/9236281.

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Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

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8

Yu, Yaoming, and Guorong Wang. "The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring." Journal of the Australian Mathematical Society 83, no. 3 (2007): 423–38. http://dx.doi.org/10.1017/s1446788700038015.

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AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

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9

Kara, Abdessalam, Néstor Thome, and Dragan Djordjevic. "Simultaneous extension of generalized BT-inverses and core-EP inverses." Filomat 38, no. 30 (2024): 10605–14. https://doi.org/10.2298/fil2430605k.

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In this paper we introduce the generalized inverse of complex square matrix with respect to other matrix having same size. Some of its representations, properties and characterizations are obtained. Also some new representation matrices of W-weighted BT-inverse and W-weighted core-EP inverse are determined as well as characterizations of generalized inverses A?, A?,W, A?, A?,W.

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10

Li, Xiezhang, and Yimin Wei. "A note on computing the generalized inverseA T,S (2)of a matrixA." International Journal of Mathematics and Mathematical Sciences 31, no. 8 (2002): 497–507. http://dx.doi.org/10.1155/s0161171202013169.

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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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11

Ji, Jun, and Yimin Wei. "The outer generalized inverse of an even-order tensor." Electronic Journal of Linear Algebra 36, no. 36 (2020): 599–615. http://dx.doi.org/10.13001/ela.2020.5011.

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Necessary and sufficient conditions for the existence of the outer inverse of a tensor with the Einstein product are studied. This generalized inverse of a tensor unifies several generalized inverses of tensors introduced recently in the literature, including the weighted Moore-Penrose, the Moore-Penrose, and the Drazin inverses. The outer inverse of a tensor is expressed through the matrix unfolding of a tensor and the tensor folding. This expression is used to find a characterization of the outer inverse through group inverses, establish the behavior of outer inverse under a small perturbation, and show the existence of a full rank factorization of a tensor and obtain the expression of the outer inverse using full rank factorization. The tensor reverse rule of the weighted Moore-Penrose and Moore-Penrose inverses is examined and equivalent conditions are also developed.

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12

Wang, Congcong, Xiaoji Liu, and Hongwei Jin. "The MP weak group inverse and its application." Filomat 36, no. 18 (2022): 6085–102. http://dx.doi.org/10.2298/fil2218085w.

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In this paper, we introduce a new generalized inverse, called MPWG inverse of a complex square matrix. We investigate characterizations, representations, and properties for this new inverse. Then, by using the core-EP decomposition, we discuss the relationships between MPWG inverse and other generalized inverses. A variant of the successive matrix squaring computational iterative scheme is given for calculating the MPWG inverse. The Cramer rule for the solution of a singular equation Ax = b is also presented. Moreover, the MPWG inverse being used in solving appropriate systems of linear equations is established. Finally, we analyze the MPWG binary relation.

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13

Yao, Jiaxuan, Xiaoji Liu, and Hongwei Jin. "Characterizations of the Generalized MPCEP Inverse of Rectangular Matrices." Journal of Applied Mathematics 2023 (March 29, 2023): 1–10. http://dx.doi.org/10.1155/2023/6235312.

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In this paper, we introduce a new generalized inverse, called the G-MPCEP inverse of a complex matrix. We investigate some characterizations, representations, and properties of this new inverse. Cramer’s rule for the solution of a singular equation A x = B is also presented. Moreover, the determinantal representations for the G-MPCEP inverse are studied. Finally, the G-MPCEP inverse being used in solving appropriate systems of linear equations is established.

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14

Jayaraman, Sachindranath. "Nonnegative generalized inverses in indefinite inner product spaces." Filomat 27, no. 4 (2013): 659–70. http://dx.doi.org/10.2298/fil1304659j.

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The aim of this article is to investigate nonnegativity of the inverse, the Moore-Penrose inverse and other generalized inverses, in the setting of indefinite inner product spaces with respect to the indefinite matrix product. We also propose and investigate generalizations of the corresponding notions of matrix monotonicity, namely, o-(rectangular) monotonicity, o-semimonotonicity and ?-weak monotonicity and its interplay with nonnegativity of various generalized inverses in the same setting.

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15

Zhang, Kaiyue, Xiaoji Liu, and Hongwei Jin. "1WG inverse of square matrices." Filomat 38, no. 12 (2024): 4225–39. https://doi.org/10.2298/fil2412225z.

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In this paper, we introduce a new generalized inverse, which is called 1WG inverse of complex square matrices. We investigate the existence and uniqueness for the 1WG inverse and give some characterizations, representations, and properties of it. Next, by using the core-EP decomposition, we discuss the relationships between the 1WG inverse and other generalized inverses. Successive matrix squaring algorithm is considered for calculating the 1WG inverse. In the end, we present a binary relation for the 1WG inverse.

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16

Röbenack, Klaus, and Kurt Reinschke. "On generalized inverses of singular matrix pencils." International Journal of Applied Mathematics and Computer Science 21, no. 1 (2011): 161–72. http://dx.doi.org/10.2478/v10006-011-0012-3.

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On generalized inverses of singular matrix pencilsLinear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

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17

Wu, Cang, and Jianlong Chen. "Minimal rank weak Drazin inverses: a class of outer inverses with prescribed range." Electronic Journal of Linear Algebra 39 (February 9, 2023): 1–16. http://dx.doi.org/10.13001/ela.2023.7359.

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For any square matrix $A$, it is proved that minimal rank weak Drazin inverses (Campbell and Meyer, 1978) of $A$ coincide with outer inverses of $A$ with range $\mathcal{R}(A^{k})$, where $k$ is the index of $A$. It is shown that the minimal rank weak Drazin inverse behaves very much like the Drazin inverse, and many generalized inverses such as the core-EP inverse and the DMP inverse are its special cases.

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18

Mosić, Dijana. "Minimal rank weighted weak Drazin inverses." Electronic Journal of Linear Algebra 40 (October 22, 2024): 714–28. http://dx.doi.org/10.13001/ela.2024.8825.

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The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

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19

Artidiello, Santiago, Alicia Cordero, Juan R. Torregrosa, and María P. Vassileva. "Generalized Inverses Estimations by Means of Iterative Methods with Memory." Mathematics 8, no. 1 (2019): 2. http://dx.doi.org/10.3390/math8010002.

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A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

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20

Mosić, D., P. S. Stanimirović, and L. A. Kazakovtsev. "The $ m $-weak group inverse for rectangular matrices." Electronic Research Archive 32, no. 3 (2024): 1822–43. http://dx.doi.org/10.3934/era.2024083.

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<abstract><p>An extension of the $ m $-weak group inverse (or $ m $-WGI) on the set of rectangular matrices is provided to solve some systems of matrix equations. The extension is termed as the $ W $-weighted $ m $-WGI (or $ W $-$ m $-WGI). The $ W $-$ m $-WGI presents a new, wider class of generalized inverses which involves some already defined generalized inverses, such as the $ m $-WGI, $ W $-weighted weak group, and $ W $-weighted Drazin inverse. Basic properties and diverse characterizations are proved for $ W $-$ m $-WGI. Several expressions for computing $ W $-$ m $-WGI are proposed in terms of known generalized inverses and projectors, as well as its limit and integral representations. The $ W $-$ m $-WGI class is utilized to solve some linear matrix equations and express their general solutions. Some new properties of the weighted generalized group inverse and recognized properties of the $ W $-weighted Drazin inverse are obtained as corollaries. Numerical and symbolic test examples are presented to verify the obtained results.</p></abstract>

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21

Zhang, Daochang, and Xiankun Du. "Representations for the Drazin inverse of a modified matrix." Filomat 29, no. 4 (2015): 853–63. http://dx.doi.org/10.2298/fil1504853z.

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In this paper expressions for the Drazin inverse of a modified matrix A - CDdB are presented in terms of the Drazin inverses of A and the generalized Schur complement D - BAdC under weaker restrictions. Our results generalize and unify several results in the literature and the Sherman-Morrison- Woodbury formula.

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22

Nie, Wenqing, and Junchao Wei. "SEP matrices and solution of matrix equations." Filomat 36, no. 13 (2022): 4591–98. http://dx.doi.org/10.2298/fil2213591n.

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This paper mainly introduces some properties of several generalized inverses of matrices, especially some equivalent characteristics of generalized inverses of matrices, specifically by constructing some specific matrix equations and discussing whether these matrix equations have solutions in a given set to determine whether a group invertible matrix is some generalized inverse of matrices.

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23

Liu, Na, and Hongxing Wang. "The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem." Journal of Mathematics 2021 (December 27, 2021): 1–10. http://dx.doi.org/10.1155/2021/4952943.

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Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.

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24

Jiang, Tianhe, Hongxing Wang, and Yimin Wei. "Perturbation of Dual Group Generalized Inverse and Group Inverse." Symmetry 16, no. 9 (2024): 1103. http://dx.doi.org/10.3390/sym16091103.

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Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A^, (which is usually represented in the form A^=A+εA0, A and A0 are, respectively, the standard part and the infinitesimal part of A^) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0.

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25

Gao, Yuefeng, Jianlong Chen, Pedro Patrício, and Dingguo Wang. "The pseudo core inverse of a companion matrix." Studia Scientiarum Mathematicarum Hungarica 55, no. 3 (2018): 407–20. http://dx.doi.org/10.1556/012.2018.55.3.1398.

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The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

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26

Stanimirović, Predrag S., Miroslav Ćirić, Igor Stojanović, and Dimitrios Gerontitis. "Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses." Complexity 2017 (2017): 1–27. http://dx.doi.org/10.1155/2017/6429725.

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Conditions for the existence and representations of 2-, 1-, and 1,2-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

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27

Sheibani, Abdolyousefi. "P-Hirano inverses in rings." Filomat 34, no. 13 (2020): 4473–82. http://dx.doi.org/10.2298/fil2013473s.

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We introduce and study a new class of generalized inverses in rings. An element a in a ring R has p-Hirano inverse if there exists b ? R such that bab = b,b ? comm2(a),(a2-ab)k ? J(R) for some k ? N. We prove that a ? R has p-Hirano inverse if and only if there exists p = p2 ? comm2(a) such that (a2-p)k ? J(R) for some k ? N. Multiplicative and additive properties for such generalized inverses are thereby obtained. We then completely determine when a 2 x 2 matrix over local rings has p-Hirano inverse.

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28

Prasad, K. Manjunatha, and M. David Raj. "Bordering method to compute Core-EP inverse." Special Matrices 6, no. 1 (2018): 193–200. http://dx.doi.org/10.1515/spma-2018-0016.

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Abstract Following the work of Kentaro Nomakuchi[10] and Manjunatha Prasad et.al., [7] which relate various generalized inverses of a given matrix with suitable bordering,we describe the explicit bordering required to obtain core-EP inverse, core-EP generalized inverse. The main result of the paper also leads to provide a characterization of Drazin index in terms of bordering.

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29

Zhang, Daochang, Dijana Mosic, and Jianping Hu. "The Drazin inverse matrix modification formulae with Peirce corners." Filomat 35, no. 8 (2021): 2605–16. http://dx.doi.org/10.2298/fil2108605z.

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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.

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30

Chen, Huanyin, and Marjan Sheibani. "Expressions for the g-Drazin inverse in a Banach algebra." Filomat 34, no. 11 (2020): 3845–54. http://dx.doi.org/10.2298/fil2011845c.

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We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a,b ? Ad. If ab = ?a?bab? for a nonzero complex number ?, then a + b ? Ad. The explicit representation of (a + b)d is presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.

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31

Sheng, Xingping. "Algebraic Perturbation Theorems of Core Inverse A # and Core-EP Inverse A †." Journal of Mathematics 2023 (February 24, 2023): 1–10. http://dx.doi.org/10.1155/2023/4110507.

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In this paper, the algebraic perturbation theorems of the core inverse A # and core-EP inverse A † are discussed for a square singular matrix A with different indices, and the expressions of the algebraic perturbation for these two new generalized inverses are presented. As their applications, some properties of the core inverse A # and core-EP inverse A † are proven again by using the expressions of their algebraic perturbation. In the last section, two numerical examples are considered to demonstrate the main results.

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32

Qin, Yingying, and Zhiping Xiong. "A Note on the Reverse Order Law for g-Inverse of Operator Product." Axioms 11, no. 5 (2022): 226. http://dx.doi.org/10.3390/axioms11050226.

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The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of the operator product. In this paper, we study the reverse order law for the g-inverse of an operator product T1T2T3 using the technique of matrix form of bounded linear operators. In particular, some necessary and sufficient conditions for the inclusion T3{1}T2{1}T1{1} ⊆ (T1T2T3){1} is presented. Moreover, some finite dimensional results are extended to infinite dimensional settings.

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33

Schuster, Gerard T., Daniel P. Johnson, and Daniel J. Trentman. "Numerical verification and extension of an analytic generalized inverse for common‐depth‐point and vertical‐seismic‐profile traveltime equations." GEOPHYSICS 53, no. 3 (1988): 326–33. http://dx.doi.org/10.1190/1.1442466.

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This paper empirically establishes the range of validity for an analytic generalized inverse (assuming negligible ray bending) associated with one‐dimensional vertical seismic profiles (VSP) or common‐depth‐point (CDP) traveltime equations. Computer tests show that the analytic inverse closely predicts the condition number of [Formula: see text] and roughly predicts some features of the unit covariance matrix for a source offset‐to‐well depth ratio less than 1. The analytic inverse is invalid for offset‐to‐depth ratios greater than 1; i.e., when ray bending becomes severe enough to violate the assumption of negligible ray bending. To overcome the restriction of negligible ray bending, we extend the analytic inverse to traveltime equations which honor Snell’s law. Computer tests show this extended inverse is a good approximation to the actual generalized inverse. The extended inverse appears to be valid for any practical source‐receiver offset and any layered velocity structure. The important implication is that, prior to their execution, VSP or CDP experiments (over approximately one‐dimensional structures) can now be designed for optimal velocity resolution. No numerical inverses need to be computed and the condition number, covariance matrix, resolution matrix, and inverse matrix are closely approximated by analytic formulas. This development promises to allow accurate and efficient velocity analysis of VSP data or CDP gathers by least‐squares traveltime inversion.

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34

Mardiyana, Mardiyana, Na'imah Hijriati, and Thresye Thresye. "INVERS TERGENERALISASI MOORE PENROSE." EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN 15, no. 2 (2022): 78. http://dx.doi.org/10.20527/epsilon.v15i2.3667.

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The generalized inverse is a concept for determining the inverse of a singular matrix and and matrix which has the characteristic of the inverse matrix. There are several types of generalized inverse, one of which is the Moore-Penrose inverse. The matrix is called Moore Penrose inverse of a matrix if it satisfies the four penrose equations and is denoted by . Furthermore, if the matrix satisfies only the first two equations of the Moore-Penrose inverse and , then is called the group inverse of and is denoted by . The purpose of this research was to determine the group inverse of a non-diagonalizable square matrix using Jordan’s canonical form and Moore Penrose’s inverse of a singular matrix, also a non-square matrix using the Singular Value Decomposition (SVD) method. The results of this study are the sufficient condition for a matrix to have a group inverse, i.e., a matrix has an index of 1 if and only if the product of two matrices forming is a full rank factorization and is invertible. Whereas for a singular matrix and a non-square , the Moore-Penrose inverse can be determined using Singular Value Decomposition (SVD). Keywords: generalized matrix inverse, Moore Penrose inverse, group inverse, Jordan canonical form, Singular Value Decomposition.

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35

Zhang, Daochang, Dijana Mosic, and Predrag Stanimirovic. "Formulae for anti-triangular block matrices which include the Drazin inverse." Applicable Analysis and Discrete Mathematics, no. 00 (2024): 22. http://dx.doi.org/10.2298/aadm230418022z.

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The expressions for the Drazin inverse of two kinds of anti-triangular block matrices are developed under new and weaker assumptions relative to those already used recently in this subject. Applying our results concerning the Drazin inverse and anti-triangular block matrices, we propose some characterizations and representations of the Drazin inverse of a 2?2 block matrix. In this way, we expand some notable achievements in characterizing and representing generalized inverses of partitioned matrices.

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36

Ke, Yuanyuan, Zhou Wang, and Jianlong Chen. "The (b,c)-inverse for products and lower triangular matrices." Journal of Algebra and Its Applications 16, no. 12 (2017): 1750222. http://dx.doi.org/10.1142/s021949881750222x.

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Let [Formula: see text] be a semigroup and [Formula: see text]. The concept of [Formula: see text]-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the [Formula: see text]-inverse. In this paper, a new relationship between the [Formula: see text]-inverse and the Bott–Duffin [Formula: see text]-inverse is established. The relations between the [Formula: see text]-inverse of [Formula: see text] and certain classes of generalized inverses of [Formula: see text] and [Formula: see text], and the [Formula: see text]-inverse of [Formula: see text] are characterized for some [Formula: see text], where [Formula: see text]. Necessary and sufficient conditions for the existence of the [Formula: see text]-inverse of a lower triangular matrix over an associative ring [Formula: see text] are also given, and its expression is derived, where [Formula: see text] are regular triangular matrices.

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37

Zhang, Daochang, and Dijana Mosic. "Explicit formulae for the generalized drazin inverse of block matrices over a Banach algebra." Filomat 32, no. 17 (2018): 5907–17. http://dx.doi.org/10.2298/fil1817907z.

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In this paper, we give expressions for the generalized Drazin inverse of a (2,2,0) block matrix over a Banach algebra under certain circumstances, utilizing which we derive the generalized Drazin inverse of a 2x2 block matrix in a Banach algebra under weaker restrictions. Our results generalize and unify several results in the literature.

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Ariyanti, Gregoria, Ana Easti Rahayu Maya Sari, and Christina Manurung. "Notes on matrix inverse over min-plus algebra." Edelweiss Applied Science and Technology 8, no. 6 (2024): 9544–54. https://doi.org/10.55214/25768484.v8i6.4034.

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One of the semiring structures is the max-plus algebra, a set with entries equipped with the operation , which represents the maximum value, and the operation , which means addition. Another semiring structure is the min-plus algebra, a set with entries equipped with the operation , representing the minimum value, and the operation , which means addition. Matrices over min-plus algebras can have inverses determined by certain conditions. The general inverse type can define the inverse of matrices over min-plus algebras. In this paper, we will develop the characteristics of general inverse matrices over min-plus algebras. The research method used is the literature study method sourced from books and journal articles. The main result of this study is that the generalized inverse of the matrix can be obtained by determining the matrix with entry which satisfies .

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39

Mosić, Dijana. "Representations for the image-kernel (p,q)-inverses of block matrices in rings." Georgian Mathematical Journal 27, no. 2 (2020): 297–305. http://dx.doi.org/10.1515/gmj-2018-0042.

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AbstractWe present the conditions for a block matrix of a ring to have the image-kernel{(p,q)}-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel{(p,q)}-inverse in a ring with involution are investigated too.

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40

Bogveradze, G., and L. P. Castro. "Invertibility of matrix Wiener-Hopf plus Hankel operators withAPWFourier symbols." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–12. http://dx.doi.org/10.1155/ijmms/2006/38152.

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A characterization of the invertibility of a class of matrix Wiener-Hopf plus Hankel operators is obtained based on a factorization of the Fourier symbols which belong to the Wiener subclass of the almost periodic matrix functions. Additionally, a representation of the inverse, lateral inverses, and generalized inverses is presented for each corresponding possible case.

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41

Zhu, Lanping, Changpeng Zhu, and Qianglian Huang. "On the uniform boundedness and convergence of generalized, Moore-Penrose and group inverses." Filomat 31, no. 19 (2017): 5993–6003. http://dx.doi.org/10.2298/fil1719993z.

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This paper concerns the relationship between uniform boundedness and convergence of various generalized inverses. Using the stable perturbation for generalized inverse and the gap between closed linear subspaces, we prove the equivalence of the uniform boundedness and convergence for generalized inverses. Based on this, we consider the cases for the Moore-Penrose inverses and group inverses. Some new and concise expressions and convergence theorems are provided. The obtained results extend and improve known ones in operator theory and matrix theory.

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42

Sun, Yunhu, and Long Wang. "A note on the Moore-Penrose inverse of block matrices." Filomat 37, no. 1 (2023): 173–77. http://dx.doi.org/10.2298/fil2301173s.

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Motivated by the representation for the Moore-Penrose inverse of the block matrix over a *-regular ring presented in [R.E. Hartwig and P. Patr?cio, When does the Moore-Penrose inverse flip? Operators and Matrices, 6(1):181-192, 2012], we show that the formula of the Moore-Penrose inverse is the same as the expression given by [Nieves Castro-Gonz?lez, Jianlong Chen and Long Wang, Further results on generalized inverses in rings with involution, Elect. J. Linear Algebra, 30:118-134, 2015].

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43

Xiao, Qi, and Jin Zhong. "Characterizations and properties of hyper-dual Moore-Penrose generalized inverse." AIMS Mathematics 9, no. 12 (2024): 35125–50. https://doi.org/10.3934/math.20241670.

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<p>In this paper, the definition of the hyper-dual Moore-Penrose generalized inverse of a hyper-dual matrix is introduced. Characterizations for the existence of the hyper-dual Moore-Penrose generalized inverse are given, and a formula for the hyper-dual Moore-Penrose generalized inverse is presented whenever it exists. Least-squares properties of the hyper-dual Moore-Penrose generalized inverse are discussed by introducing a total order of hyper-dual numbers. We also introduce the definition of a dual matrix of order $ n $. A necessary and sufficient condition for the existence of the Moore-Penrose generalized inverse of a dual matrix of order $ n $ is given.</p>

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44

Jiang, Bo, and Yongge Tian. "Invariance property of a five matrix product involving two generalized inverses." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 1 (2021): 83–92. http://dx.doi.org/10.2478/auom-2021-0006.

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Abstract Matrix expressions composed by generalized inverses can generally be written as f(A − 1, A − 2, . . ., A − k ), where A 1, A 2, . . ., A k are a family of given matrices of appropriate sizes, and (·)− denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality A 1 A − 2 A 3 A − 4 A 5 = A to always hold for all generalized inverses A − 2 and A − 4 of A 2 and A 4 through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices.

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45

Yoshida, Ikumasa, and Michihiro KITAHARA. "Inverse analysis of stochastic field by using extended generalized inverse matrix." Journal of applied mechanics 7 (2004): 185–90. http://dx.doi.org/10.2208/journalam.7.185.

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46

Tian, Yongge. "Characterizations of matrix equalities involving the sums and products of multiple matrices and their generalized inverse." Electronic Research Archive 31, no. 9 (2023): 5866–93. http://dx.doi.org/10.3934/era.2023298.

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<abstract><p>It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.</p></abstract>

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47

Cordero, Alicia, Javier G. Maimó, Juan R. Torregrosa, and María P. Vassileva. "Improving Newton–Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory." Mathematics 11, no. 14 (2023): 3161. http://dx.doi.org/10.3390/math11143161.

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Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary m×n complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.

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48

Zuo, Kezheng, Yang Chen, and Li Yuan. "Further representations and computations of the generalized Moore-Penrose inverse." AIMS Mathematics 8, no. 10 (2023): 23442–58. http://dx.doi.org/10.3934/math.20231191.

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<abstract><p>The aim of this paper is to provide new representations and computations of the generalized Moore-Penrose inverse. Based on the Moore-Penrose inverse, group inverse, Bott-Duffin inverse and certain projections, some representations for the generalized Moore-Penrose inverse are given. An equivalent condition for the continuity of the generalized Moore-Penrose inverse is proposed. Splitting methods and successive matrix squaring algorithm for computing the generalized Moore-Penrose inverse are presented.</p></abstract>

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Chen, Huanyin, and Marjan Abdolyousefi. "Additive properties of g-Drazin invertible linear operators." Filomat 36, no. 10 (2022): 3301–9. http://dx.doi.org/10.2298/fil2210301c.

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In this paper, we investigate additive properties of generalized Drazin inverse for bounded linear operators. As an application we present new conditions under which a 2 ? 2 operator matrix has g-Drazin inverse. These extend the main results of Dana and Yousefi (Int. J. Appl. Comput. Math., 4(2018), page 9), Yang and Liu (J. Comput. Appl. Math., 235(2011), 1412-1417) and Sun et al. (Filomat, 30(2016), 3377-3388).

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DAMM, TOBIAS, and HARALD K. WIMMER. "A CANCELLATION PROPERTY OF THE MOORE–PENROSE INVERSE OF TRIPLE PRODUCTS." Journal of the Australian Mathematical Society 86, no. 1 (2009): 33–44. http://dx.doi.org/10.1017/s144678870800044x.

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AbstractWe study the matrix equation C(BXC)†B=X†, where X† denotes the Moore–Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore–Penrose inverses are given.

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