Characterizations of the group invertibility of a matrix revisited

• Autorzy: Tian Yongge

Identyfikatory

DOI

10.1515/dema-2022-0171

• Języki publikacji: EN

Abstrakty

EN

A square complex matrix A is said to be group invertible if there exists a matrix X such that AXA=A, XAX=X, and AX=XA hold, and such a matrix X is called the group inverse of A . The group invertibility of a matrix is one of the fundamental concepts in the theory of generalized inverses, while group inverses of matrices have many essential applications in matrix theory and other disciplines. The purpose of this article is to reconsider the characterization problem of the group invertibility of a matrix, as well as the constructions of various algebraic equalities in relation to group invertible matrices. The coverage includes collecting and establishing a family of existing and new necessary and sufficient conditions for a matrix to be group invertible and giving many algebraic matrix equalities that involve Moore-Penrose inverses and group inverses of matrices through the skillful use of a series of highly selective formulas and facts about ranks, ranges, and generalized inverses of matrices, as well as block matrix operations.

Słowa kluczowe

EN

block matrix; group inverse; Moore-Penrose inverse; range; rank

PL

macierz blokowa; odwrotność Moore-Penrose; odległość; ranga

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 866--890
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Tian Yongge

• College of Business and Economics, Shanghai Business School, Shanghai 201400, China

Bibliografia

• [1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed., Springer, New York, 2003.
• [2] D. S. Bernstein, Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas Revised and Expanded Edition, Princeton University Press, Princeton, NJ, USA, 2018.
• [3] S. L. Campbell and C. D. Meyer Jr., Generalized Inverses of Linear Transformations, SIAM, Philadelphia, 2009.
• [4] C. Bu, J. Zhao, and J. Zheng, Group inverse for a class ×2 2 block matrices over skew fields, Appl. Math. Comput. 204 (2008), 45–49.
• [5] C. Cao, Y. Wang, and Y. Sheng, Group inverses for some ×2 2 block matrices over rings, Front. Math. China 11 (2016), 521–538.
• [6] C. Cao, H. Zhang, and Y. Ge, Further results on the group inverse of some anti-triangular block matrices, J. Appl. Math. Comput. 46 (2014), 169–179.
• [7] C. Cao, X. Zhang, and X. Tang, Reverse order law of group inverses of products of two matrices, Appl. Math. Comput. 158 (2004), 489–495.
• [8] C.-Y. Deng, Reverse order law for the group inverses, J. Anal. Math. Appl. 382 (2011), 663–671.
• [9] C.-Y. Deng, On the group invertibility of operators, Electron. J. Linear Algebra 31 (2016), 492–510.
• [10] S. K. Mitra, On group inverses and the sharp order, Linear Algebra Appl. 92 (1987), 17–37.
• [11] D. Mosić and D. S. Djordjević, The reverse order law ( ) ( )=#ab b a abb a† † † † † in rings with involution, RACSAM 109 (2015), 257–265
• [12] P. Robert, On the group inverse of a linear transformation, J. Math. Anal. Appl. 22 (1968), 658–669.
• [13] Y. Sheng, Y. Ge, H. Zhang, and C. Cao, Group inverses for a class of ×2 2 block matrices over rings, Appl. Math. Comput. 219 (2013), 9340–9346.
• [14] D. Zhang, D. Mosic, and T.-Y. Tam, On the existence of group inverses of Peirce corner matrices, Linear Algebra Appl. 582 (2019), 482–498.
• [15] P. Basavappa, On the solutions of the matrix equation ( ) ( )=∗ ∗f X X g X X, , , Canad. Math. Bull. 15 (1972), 45–49.
• [16] S. A. McCullough and L. Rodman, Hereditary classes of operators and matrices, Amer. Math. Monthly 104 (1997), 415–430.
• [17] Y. Tian, Reverse order laws for the generalized inverses of multiple matrix products, Linear Algebra Appl. 211 (1994), 85–100.
• [18] Y. Tian, Rank equalities related to outer inverses of matrices and applications, Linear Multilinear Algebra 49 (2002), 269–288.
• [19] Y. Tian, The reverse-order law ( ) ( )=AB B A ABB A† † † † † † and its equivalent equalities, J. Math. Kyoto Univ. 45 (2005), 841–850.
• [20] Y. Tian, A family of 512 reverse order laws for generalized inverses of a matrix product: a review, Heliyon 6 (2020), e04924.
• [21] Y. Tian, Some mixed-type reverse-order laws for the Moore-Penrose inverse of a triple matrix product, Rocky Mt. J. Math. 37 (2007), 1327–1347.
• [22] Y. Tian, Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product, AIMS Math. 6 (2021), 13845–13886.
• [23] R. E. Hartwig, Block generalized inverses, Arch. Rat. Mech. Anal. 61 (1976), 197–251.
• [24] R. E. Hartwig, Group inverses and Drazin inverses of bidiagonal and triangular Toeqlitz matrices, J. Austral. Math. Soc. 24 (1977), 10–34.
• [25] P. Patrício and R. E. Hartwig, The (2,2,0) group inverse problem, Appl. Math. Comput. 217 (2010), 516–520.
• [26] R. E. Hartwig and K. Spindelböck, Matrices for which ∗A and A† can commute, Linear Multilinear Algebra 14 (1983), 241–256.
• [27] J. Cen, On existence of weighted group inverse of rectangular matrix (in Chinese), Math. Numer. Sin. 29 (2007), 39–48.
• [28] Y. Chen, Existence conditions and expressions for weighted group inverses of rectangular matrices, J. Nanjing Norm. Univ. Nat. Sci. Edn. 31 (2008), no. 3, 1–5.
• [29] R. E. Cline and T. N. E. Greville, A Drazin inverse for rectangular matrices, Linear Algebra Appl. 29 (1980), 53–62.
• [30] X. Sheng and G. Chen, The computation and perturbation analysis for weighted group inverse of rectangular matrices, J. Appl. Math. Comput. 31 (2008), 33–43.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).