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Journal articles on the topic 'Inverse'

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Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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1

Sahoo, Jajati Keshari, Gayatri Maharana, Bibekananda Sitha, and Nestor Thome. "1D inverse and D1 inverse of square matrices." Miskolc Mathematical Notes 25, no. 1 (2024): 445. http://dx.doi.org/10.18514/mmn.2024.4339.

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In this paper we introduce two new classes of inverses for square matrices, which are called 1Drazin (in short, 1D) inverse and Drazin1 (in short, D1) inverse. Next, we investigated the existence and uniqueness of 1D inverse and its dual D1 inverse. Some representation and characterizations of these inverses are derived. In addition to this, we obtain some properties of 1D and D1 inverses through idempotent and binary relations.

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2

Vitoshkin, H., and A. Yu Gelfgat. "On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD." Communications in Computational Physics 14, no. 4 (October 2013): 1103–19. http://dx.doi.org/10.4208/cicp.300412.010213a.

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AbstractFactorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

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3

Jansen-Osmann, Petra, Steffen Beirle, Stefanie Richter, Jürgen Konczak, and Karl-Theodor Kalveram. "Inverse Motorische Modelle bei Kindern und Erwachsenen:." Zeitschrift für Entwicklungspsychologie und Pädagogische Psychologie 34, no. 3 (July 2002): 167–73. http://dx.doi.org/10.1026//0049-8637.34.3.167.

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Zusammenfassung. Neurobiologische Befunde zeigen, dass neuronal kodierte inverse Modelle der Biomechanik des Körpers die Basis menschlicher Bewegungssteuerung bilden. Diese Arbeit untersucht die Rolle visueller Information zur Präzisierung inverser motorischer Modelle bei Kindern und Erwachsenen. Je 8 neun- bzw. fünfjährige Kinder und 8 Erwachsene führten horizontale, zielgerichtete Unterarmbewegungen unter Variation visuellen Feedbacks durch (volles Feedback, partielles Feedback, kein Feedback). Die Bewegungen Erwachsener waren ungenauer, wenn das visuelle Feedback am Anfang und Ende der Bewegung fehlte (partielles Feedback). Das Fehlen visueller Information während der Bewegung führte bei ihnen zu keiner Vergrößerung der räumlichen Zielabweichung. Dagegen waren die Bewegungen beider Kindergruppen ungenauer, wenn visuelles Feedback während der Bewegung fehlte. Die Zielgenauigkeit verbesserte sich jedoch, wenn die Bewegungen zuvor unter visuellem Feedback gelernt wurden. Die Abhängigkeit der Kinder von visuellem Feedback bedeutet, dass sie stärker auf zentrale Regelungsprozesse angewiesen sind. Die mangelnde motorische Steuerung besonders der jüngeren Kinder hat wahrscheinlich zwei Ursachen: Erstens, eine fehlerhafte invers kinematische Transformation (von Zielkoordinaten zu Gelenkwinkeln) und zweitens, ein ungenau parametrisiertes inverses Modell.

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4

Xu, San-Zhang, Julio Benítez, Ya-Qian Wang, and Dijana Mosić. "Two Generalizations of the Core Inverse in Rings with Some Applications." Mathematics 11, no. 8 (April 12, 2023): 1822. http://dx.doi.org/10.3390/math11081822.

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In this paper, we introduce two new generalized core inverses, namely, the (p,q,m)-core inverse and the ⟨p,q,n⟩-core inverse; both extend the inverses of the ⟨i,m⟩-core inverse, the (j,m)-core inverse, the core inverse, the core-EP inverse and the DMP-inverse.

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5

Li, Tingting, Dijana Mosic, and Jianlong Chen. "The Sherman-Morrison-Woodbury formula for the generalized inverses." Filomat 36, no. 15 (2022): 5307–13. http://dx.doi.org/10.2298/fil2215307l.

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In this paper, we investigate the Sherman-Morrison-Woodbury formula for the {1}-inverses and the {2}-inverses of bounded linear operators on a Hilbert space. Some conditions are established to guarantee that (A+YGZ*)? = A? ?A?Y(G? +Z*A?Y)?Z*A? holds, where A? stands for any kind of standard inverse, {1}-inverse, {2}-inverse, Moore-Penrose inverse, Drazin inverse, group inverse, core inverse and dual core inverse of A.

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6

Zhou, Mengmeng, Jianlong Chen, and Néstor Thome. "The W-weighted Drazin-star matrix and its dual." Electronic Journal of Linear Algebra 37, no. 37 (February 3, 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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7

Udwadia, Firdaus E. "When Does a Dual Matrix Have a Dual Generalized Inverse?" Symmetry 13, no. 8 (July 30, 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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8

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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9

Ji, Jun, and Yimin Wei. "The outer generalized inverse of an even-order tensor." Electronic Journal of Linear Algebra 36, no. 36 (September 5, 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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10

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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11

Ke, Yuanyuan, Long Wang, Jiahui Liang, and Ling Shi. "Right e-core inverse and the related generalized inverses in rings." Filomat 37, no. 15 (2023): 5039–51. http://dx.doi.org/10.2298/fil2315039k.

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In this paper, some characterizations and properties of right e-core inverses by using right invertible element and {1, 3e}-inverse are investigated. Meanwhile, some characterizations for a new generalized right e-core inverse which is called right pseudo e-core inverse are also studied. The relationship between right pseudo e-core inverses and right e-core inverses are presented.

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12

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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13

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 (November 20, 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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14

Benson, Christine C., and Margaret Buerman. "The Inverse Name Game." Mathematics Teacher 101, no. 2 (September 2007): 108–12. http://dx.doi.org/10.5951/mt.101.2.0108.

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The importance of emphasizing the concept of inverse. It addresses the different names we use for inverses and suggests ways to help students see the big picture. The author ties together the use of inverses in functions, arithmetic operations, reciprocal relationships, and makes a case for instruction of the inverse as a coherent whole.

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15

Visnjic, Jelena, Ivana Stanisev, and Yuanyuan Ke. "Reverse order law and forward order law for the (b, c)-inverse." Electronic Journal of Linear Algebra 39 (July 13, 2023): 379–94. http://dx.doi.org/10.13001/ela.2023.7807.

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The reverse order law and the forward order law have been studied for various types of generalized inverses. The $(b,c)$-inverse is a generalization of some well known generalized inverses, such as the Moore-Penrose inverse, the Drazin inverse, the core inverse, etc. In this paper, the reverse order law for the $(b,c)$-inverse, in a unital ring, is investigated and an equivalent condition for this law to hold for the $(b,c)$-inverse is derived. Also, some known results on this topic are generalized. Furthermore, the forward order law for the $(b,c)$-inverse in a ring with a unity is introduced, for different choices of $b$ and $c$. Moreover, as corollaries of obtained results, equivalent conditions for the reverse order law and the forward order law for the inverse along an element are derived.

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16

Marzuki, Corry Corazon, and Yulia Rosita. "Generalized Inverse Pada Matriks Atas Zn." Jurnal Sains Matematika dan Statistika 1, no. 2 (August 10, 2016): 1. http://dx.doi.org/10.24014/jsms.v1i2.1953.

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Suatu matriks mempunyai invers apabila matriks tersebut non-singular dan bujur sangkar. Namun, apabila matriks tersebut singular atau tidak bujur sangkar, inversnya masih dapat ditentukan dengan generalized inverse. Pada tugas akhir ini dibahas bagaimana menentukan generalized inverse pada matriks atas menggunakan aturan algoritma dan aturan pendiagonalan matriks. Berdasarkan pembahasan pada tugas akhir ini dapat disimpulkan bahwa apabila merupakan bilangan prima makaadalah lapangan dan matriks atas pasti mempunyai generalized inverse. Namun apabila bukan bilangan prima maka adalah ring komutatif dengan elemen satuan dan matriks atas mempunyai generalized inverse apabila dalam proses pengerjaan tidak dibutuhkan invers dari suatu elemen atas yang tidak mempunyai invers terhadap perkalian. Adapun generalized inverse yang diperoleh adalah tidak tunggal.

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17

Purba, Wahyu Aditya Suranta, M. Hasbi, Usman Usman, and RM Bambang. "Pemahaman Siswa Sekolah Menengah Atas tentang Konsep Fungsi Invers." Jurnal Ilmiah Soulmath : Jurnal Edukasi Pendidikan Matematika 12, no. 1 (January 25, 2024): 55–74. http://dx.doi.org/10.25139/smj.v12i1.6835.

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Abstract Mathematical understanding is a person's capacity to recognize mathematical information and how this information is used as a strategy for solving an existing problem to achieve ease in obtaining the truth. However, students are still found who lack an understanding of inverse functions. This is very unfortunate, even though the inverse function is one of the materials that students must understand. This research aims to analyze students' mathematical understanding of inverse function material. This type of research is descriptive qualitative. The research subjects were 28 class XI students at one of Banda Aceh's high schools. Data collection used two instruments, namely a mathematical understanding test, and an interview guide. The triangulation used is time triangulation. The indicators tested on the test subjects were 1) restating the definition of the inverse function; 2) identifying functions that have an inverse and do not have an inverse; 3) showing examples and non-examples of functions that have an inverse; and 4) apply procedures and algorithms in solving inverse function problems. Based on the results of the analysis, show that a small number of students fulfill all the indicators of understanding in solving inverse function questions. Most students can apply the procedure but are unable to relate the answer to the concept of inverse functions. Keywords: Inverse functions; Comprehension indicators; Mathematical understanding. Abstrak Pemahaman matematis adalah kapasitas seseorang untuk mengenali informasi matematika dan bagaimana informasi tersebut digunakan sebagai strategi penyelesaian suatu permasalahan yang ada sehingga tercapainya kemudahan dalam memperoleh kebenaran. Namun faktanya masih ditemukan siswa yang kurang memiliki pemahaman fungsi invers. Hal ini sangat disayangkan, padahal fungsi invers merupakan salah satu materi yang harus dipahami oleh siswa. Penelitian ini bertujuan untuk menganalisis bagaimana pemahaman matematis siswa pada materi fungsi invers. Jenis penelitian ini adalah deskriptif kualitatif. subjek penelitian yaitu 28 siswa kelas XI di salah satu SMA Banda Aceh. Pengumpulan data menggunakan dua instrument yaitu tes pemahaman matematis dan panduan wawancara. Triangulasi yang digunakan adalah triangulasi waktu. Indikator yang diuji kepada subjek tes yaitu 1) menyatakan ulang definisi fungsi invers; 2) mengidentifikasikan fungsi yang memiliki invers dan tidak memiliki invers; 3) menunjukkan contoh dan non-contoh fungsi yang memiliki invers; dan 4) menerapkan prosedur dan algoritma dalam menyelesaikan soal fungsi invers. Berdasarkan hasil analisis menunjukkan sebagian kecil siswa memenuhi keseluruhan indikator pemahaman dalam menyelesaikan soal fungsi invers. Sebagian besar siswa mampu menerapkan prosedur namun tidak mampu mengaitkan jawaban dengan konsep dari fungsi invers. Kata Kunci: Fungsi invers; Indikator Pemahaman; Pemahaman Matematis.

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18

Gu, Caixing, Heidi Keas, and Robert Lee. "The n-inverses of a matrix." Filomat 31, no. 12 (2017): 3801–13. http://dx.doi.org/10.2298/fil1712801g.

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The concept of a left n-inverse of a bounded linear operator on a complex Banach space was introduced recently. Previously, there have been results on products and tensor products of left n-inverses, and the representation of left n-inverses as the sum of left inverses and nilpotent operators was being discussed. In this paper, we give a spectral characterization of the left n-inverses of a finite (square) matrix. We also show that a left n-inverse of a matrix T is the sum of the inverse of T and two nilpotent matrices.

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19

Mosic, Dijana, Honglin Zou, and Long Wang. "Extension of the generalized n-strong Drazin inverse." Filomat 37, no. 23 (2023): 7781–90. http://dx.doi.org/10.2298/fil2323781m.

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The aim of this paper is to present an extension of the generalized n-strong Drazin inverse for Banach algebra elements using a g-Drazin invertible element rather than a quasinilpotent element in the definition of the generalized n-strong Drazin inverse. Thus, we introduce a new class of generalized inverses which is a wider class than the classes of the generalized n-strong Drazin inverse and the extended generalized strong Drazin inverses. We prove a number of characterizations for this new inverse and some of them are based on idempotents and tripotents. Several generalizations of Cline?s formula are investigated for the extension of the generalized n-strong Drazin inverse.

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20

Rao, P. S. S. N. V. P. "On Group Inverses of Matrices with Order and Rank Perturbations." Calcutta Statistical Association Bulletin 44, no. 3-4 (September 1994): 209–22. http://dx.doi.org/10.1177/0008068319940308.

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Group inverse of a square matrix A exists if and only if rank of A is equal to rank of A2. Group inverses have many applications, prominent among them is in the analysis of finite Markov chains discussed by Meyer (1982). In this note necessary and sufficient conditions for the existence of group inverses of bordered matrix, [Formula: see text] are obtained and expressions for the group inverses in terms of group inverse of A are given, whenever they exist. Also necessary and sufficient condition for the existence of group inverse of A in terms of group inverse of B and C are given. An application to perturbation in Markov chains is illustrated.

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21

Chen, Huanyin, and Marjan Sheibani. "Generalized Hirano inverses in Banach algebras." Filomat 33, no. 19 (2019): 6239–49. http://dx.doi.org/10.2298/fil1919239c.

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Let A be a Banach algebra. An element a ? A has generalized Hirano inverse if there exists b ? A such that b = bab, ab = ba, a2-ab ? Aqnil. We prove that a ? A has generalized Hirano inverse if and only if a - a3 ? Aqnil, if and only if a is the sum of a tripotent and a quasinilpotent that commute. The Cline?s formula for generalized Hirano inverses is thereby obtained. Let a, b ? A have generalized Hirano inverses. If a2b = aba and b2a = bab, we prove that a + b has generalized Hirano inverse if and only if 1 + adb has generalized Hirano inverse. The generalized Hirano inverses of operator matrices on Banach spaces are also studied.

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22

Pablos Romo, Fernando. "Generalized inverses of bounded finite potent operators on Hilbert spaces." Filomat 36, no. 18 (2022): 6139–58. http://dx.doi.org/10.2298/fil2218139p.

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The aim of this work is to prove the existence and uniqueness of the Drazin inverse and the DMP inverses of a bounded finite potent endomorphism. In particular, we give the main properties of these generalized inverses, we offer their relationships with the adjoint operator, we study their spectrum, we compute the respective traces and determinants and we relate the Drazin inverse of a bounded finite potent operator with classical definitions of this generalized inverse. Moreover, different properties of the Moore-Penrose inverse of a bounded operator are studied.

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23

Benitez, Julio, and Enrico Boasso. "The inverse along an element in rings." Electronic Journal of Linear Algebra 31 (February 5, 2016): 572–92. http://dx.doi.org/10.13001/1081-3810.3113.

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Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

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24

Sitha, Bibekananda, Jajati Sahoo, and Ratikanta Behera. "Characterization of weighted (b,c) inverse of an element in a ring." Filomat 36, no. 14 (2022): 4629–44. http://dx.doi.org/10.2298/fil2214629s.

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The notion of the weighted (b,c)-inverse of an element in rings were introduced very recently. In this paper, we further elaborate on this theory by establishing a few characterizations of this inverse and their relationships with other (v,w)-weighted (b,c)-inverses. We discuss a few necessary and sufficient conditions for the existence of the hybrid (v,w)-weighted (b,c)-inverse and the annihilator (v,w)-weighted (b, c)-inverse of an element in a ring. In addition, we explore a few sufficient conditions for the reverse-order law of the annihilator (v,w)-weighted (b,c)-inverses.

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25

Jin, Hongwei, Mengyu He, and Yuzhen Wang. "The expressions of the generalized inverses of the block tensor via the C-product." Filomat 37, no. 26 (2023): 8909–26. http://dx.doi.org/10.2298/fil2326909j.

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In this paper, we present the expressions of the generalized inverses of the third-order 2 ? 2 block tensor under the C-Product. Firstly, we give the necessary and sufficient conditions to present some generalized inverses and the Moore-Penrose inverse of the block tensor in Banachiewicz-Schur forms. Next, some results are generalized to the group inverse and the Drazin inverse. Moreover, equivalent conditions for the existence as well as the expressions for the core inverse of the block tensor are obtained. Finally, the results are applied to express the quotient property and the first Sylsvester identity of tensors.

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26

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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27

Wu, Zhenying, and Qingping Zeng. "Extensions of Cline’s formula for some new generalized inverses." Filomat 35, no. 2 (2021): 477–83. http://dx.doi.org/10.2298/fil2102477w.

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Let a, b, c, d be elements in a unital associative ring R. In this note, we generalize Cline?s formula for some new generalized inverses such as strong Drazin inverse, generalized strong Drazin inverse, Hirano inverse and generalized Hirano inverse to the case when acd = dbd and dba = aca. As a particular case, some recent results are recovered.

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28

Kyrchei, Ivan I. "Determinantal Representations of the Core Inverse and Its Generalizations with Applications." Journal of Mathematics 2019 (October 1, 2019): 1–13. http://dx.doi.org/10.1155/2019/1631979.

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In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

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29

Mosic, Dijana, Predrag Stanimirovic, and Miroslav Ciric. "Extensions of G-outer inverses." Filomat 37, no. 22 (2023): 7407–29. http://dx.doi.org/10.2298/fil2322407m.

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Our first objective is to present equivalent conditions for the solvability of the system of matrix equations ADA = A, D= B and CAD = C, where D is unknown, A, B,C are of appropriate dimensions, and to obtain its general solution in terms of appropriate inner inverses. Our leading idea is to find characterizations and representations of a subclass of inner inverses that satisfy some properties of outer inverses. A G-(B,C) inverse of A is defined as a solution of this matrix system. In this way, G-(B,C) inverses are defined and investigated as an extension of G-outer inverses. One-sided versions of G-(B,C) inverse are introduced as weaker kinds of G-(B,C) inverses and generalizations of one-sided versions of G-outer inverse. Applying the G-(B,C) inverse and its one-sided versions, we propose three new partial orders on the set of complex matrices. These new partial orders extend the concepts of G-outer (T, S)-partial order and one-sided G-outer (T, S)-partial orders.

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30

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 (December 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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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

Chen, Saijie, Yayuan Zhao, Lanping Zhu, and Qianglian Huang. "The continuity and the simplest possible expression of inner inverses of linear operators in Banach space." Filomat 35, no. 4 (2021): 1241–51. http://dx.doi.org/10.2298/fil2104241c.

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The main topic of this paper is the relationship between the continuity and the simplest possible expression of inner inverses. We first provide some new characterizations for the simplest possible expression to be an inner inverse of the perturbed operator. Then we obtain the equivalence conditions on the continuity of the inner inverse. Furthermore, we prove that if Tn ? T and the sequence of inner inverses {T?n} is convergent, then T is inner invertible and we can find a succinct expression of the inner inverse of Tn, which converge to any given inner inverse T?. This is very useful and convenient in applications.

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33

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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34

Benitez, Julio, Enrico Boasso, and Hongwei Jin. "ON ONE-SIDED (B;C)-INVERSES OF ARBITRARY MATRICES." Electronic Journal of Linear Algebra 32 (February 6, 2017): 391–422. http://dx.doi.org/10.13001/1081-3810.3487.

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In this article, one-sided $(b, c)$-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In addition, the $(b, c)$-inverse and the inverse along an element will be also researched in the context of rectangular matrices.

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35

Kyrchei, Ivan I. "Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses." Journal of Mathematics 2020 (May 31, 2020): 1–12. http://dx.doi.org/10.1155/2020/9816038.

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In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. A numerical example to illustrate the main result is given.

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36

Zeng, Qingping, Zhenying Wu, and Yongxian Wen. "New extensions of Cline’s formula for generalized inverses." Filomat 31, no. 7 (2017): 1973–80. http://dx.doi.org/10.2298/fil1707973z.

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In this paper, Cline?s formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when ( acd = dbd dba = aca. Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

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37

González, N. Castro, and J. J. Koliha. "New additive results for the g-Drazin inverse." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 6 (December 2004): 1085–97. http://dx.doi.org/10.1017/s0308210500003632.

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This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

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38

Soleymani, F., M. Sharifi, and S. Shateyi. "Approximating the Inverse of a Square Matrix with Application in Computation of the Moore-Penrose Inverse." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/731562.

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This paper presents a computational iterative method to find approximate inverses for the inverse of matrices. Analysis of convergence reveals that the method reaches ninth-order convergence. The extension of the proposed iterative method for computing Moore-Penrose inverse is furnished. Numerical results including the comparisons with different existing methods of the same type in the literature will also be presented to manifest the superiority of the new algorithm in finding approximate inverses.

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39

Gao, Yuefeng, Jianlong Chen, Pedro Patrício, and Dingguo Wang. "The pseudo core inverse of a companion matrix." Studia Scientiarum Mathematicarum Hungarica 55, no. 3 (September 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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40

Wang, Hongxing, and Jianlong Chen. "Weak group inverse." Open Mathematics 16, no. 1 (October 31, 2018): 1218–32. http://dx.doi.org/10.1515/math-2018-0100.

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AbstractIn this paper, we introduce the weak group inverse (called as the WG inverse in the present paper) for square complex matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. The paper ends with a characterization of the core EP order using WG inverses.

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41

Wallace, Edward C. "Investigations Involving Involutions." Mathematics Teacher 81, no. 7 (October 1988): 578–79. http://dx.doi.org/10.5951/mt.81.7.0578.

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Many second-year algebra textbooks include a discussion of functions and their inverses prior to introducing logarithms and the inverse trigonometric functions. Since these functions and some others can be approached neatly as inverses of more familiar functions, a good understanding of the notion of inverse allows students to understand these topics better.

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42

Zhao, Xiangui. "Jacobson’s Lemma via Gröbner-Shirshov Bases." Algebra Colloquium 24, no. 02 (April 23, 2017): 309–14. http://dx.doi.org/10.1142/s1005386717000189.

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Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], if 1− ab is invertible then so is 1 − ba. Jacobson’s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.

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43

Djordjevic, Dragan S. "Generalized Inverses." Geometry, Integrability and Quantization 22 (2021): 13–32. http://dx.doi.org/10.7546/giq-22-2021-13-32.

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44

Ke, Yuanyuan, Yuefeng Gao, and Jianlong Chen. "Representations of the (b,c)-inverses in rings with involution." Filomat 31, no. 9 (2017): 2867–75. http://dx.doi.org/10.2298/fil1709867k.

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Let R be a ring and b,c ? R. The concept of (b,c)-inverses was introduced by Drazin in 2012. In this paper, the existence and the expression of the (b,c)-inverse in a ring with an involution are investigated. A new representation of the (b,c)-inverse based on the group inverse is also presented.

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45

Gao, Jiale, Kezheng Zuo, Qingwen Wang, and Jiabao Wu. "Further characterizations and representations of the Minkowski inverse in Minkowski space." AIMS Mathematics 8, no. 10 (2023): 23403–26. http://dx.doi.org/10.3934/math.20231189.

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<abstract><p>This paper serves to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of $ \{1, 3^{\mathfrak{m}}\} $-, $ \{1, 2, 3^{\mathfrak{m}}\} $-, $ \{1, 4^{\mathfrak{m}}\} $- and $ \{1, 2, 4^{\mathfrak{m}}\} $-inverses are given in order to represent the Minkowski inverse. Second, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Third, using the Hartwig-Spindelböck decomposition, we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.</p></abstract>

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46

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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47

Chen, Huanyin, and Tugce Calci. "ps-drazin inverses in banach algebras." Filomat 33, no. 7 (2019): 2125–33. http://dx.doi.org/10.2298/fil1907125c.

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An element a in a Banach algebra A has ps-Drazin inverse if there exists p2 = p ? comm2(a) such that (a - p)k ? J(A) for some k ? N. Let A be a Banach algebra, and let a,b ? A have ps-Drazin inverses. If a2b = aba and b2a = bab, we prove that 1. ab ? A has ps-Drazin inverse. 2. a + b ? A has ps-Drazin inverse if and only if 1 + adb ? A has ps-Drazin inverse. As applications, we present various conditions under which a 2 x 2 matrix over a Banach algebra has ps-Drazin inverse.

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48

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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49

Stanimirovic, Predrag, Xue-Zhong Wang, and Haifeng Ma. "Complex ZNN for computing time-varying weighted pseudo-inverses." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 131–64. http://dx.doi.org/10.2298/aadm170628019s.

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We classify, extend and unify various generalizations of weighted Moore-Penrose inverses in indefinite inner product spaces. New kinds of generalized inverses are introduced for this purpose. These generalized inverses are included in the more general class called as the weighted indefinite pseudoinverses (WIPI), which represents an extension of the Minkowski inverse (MI), the weighted Minkowski inverse (WMI), and the generalized weighted Moore- Penrose (GWM-P) inverse. The WIPI generalized inverses are introduced on the basis of two Hermitian invertible matrices and two Hermitian involuntary matrices and represented as particular outer inverses with prescribed ranges and null spaces, in terms of appropriate full-rank and limiting representations. Application of introduced generalized inverses in solving some indefinite least squares problems is considered. New Zeroing Neural Network (ZNN) models for computing the WIPI are developed using derived full-rank and limiting representations. The convergence behavior of the proposed ZNN models is investigated. Numerical simulation results are presented.

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50

Wang, Long, Dijana Mosić, and Yuefeng Gao. "Right core inverse and the related generalized inverses." Communications in Algebra 47, no. 11 (April 6, 2019): 4749–62. http://dx.doi.org/10.1080/00927872.2019.1596275.

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