Annals of Functional Analysis

Rank equalities for Moore-Penrose inverse and Drazin‎ ‎inverse over quaternion

Huasheng Zhang

Full-text: Open access


‎In this paper‎, ‎we consider the ranks of four real‎ ‎matrices $G_{i}(i=0,1,2,3)$ in $M^{\dagger},$ where $M=M_{0}+M_{1}%‎ ‎i+M_{2}j+M_{3}k$ is an arbitrary quaternion matrix‎, ‎and $M^{\dagger}%‎ ‎=G_{0}+G_{1}i+G_{2}j+G_{3}k$ is the Moore-Penrose inverse of $M$‎. ‎Similarly‎, ‎the ranks of four real matrices in Drazin inverse of a‎ ‎quaternion matrix are also presented‎. ‎As applications‎, ‎the necessary‎ ‎and sufficient conditions for $M^{\dagger}$ is pure real or pure‎ ‎imaginary Moore-Penrose inverse and $N^{D}$ is pure real or pure‎ ‎imaginary Drazin inverse are presented‎, ‎respectively‎.

Article information

Ann. Funct. Anal., Volume 3, Number 2 (2012), 115-127.

First available in Project Euclid: 12 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A03: Vector spaces, linear dependence, rank
Secondary: 15A09‎ ‎15A24‎ ‎15A33

Moore-penrose inverse ‎rank‎ ‎quaternion matrix ‎Drazin‎ ‎inverse


Zhang, Huasheng. Rank equalities for Moore-Penrose inverse and Drazin‎ ‎inverse over quaternion. Ann. Funct. Anal. 3 (2012), no. 2, 115--127. doi:10.15352/afa/1399899936.

Export citation


  • S. L. Campbell and C.D. Meyer, Generalized inverse of linear transformations, Corrected reprint of the 1979 original. Dover Publications, Inc., New York, 1991.
  • A. Ben-Israel and T. N. E. Greville, Generalized inverses: Theory and Applications, second ed., Springer, New York, 2003.
  • C. H. Hung and T.L. Markham, The Moore-Penrose inverse of a partitioned matrix, Linear Algebra Appl. 11 (1975), 73–86.
  • C.D. Meyer Jr., Generalized inverses and ranks of block matrices, SIAM J. Appl. Math. 25 (1973), 597–602.
  • J. Miao, General expression for Moore-Penrose invers of a $2\times2$ block matrix, Linear Algebra Appl. 151 (1990) 1–15.
  • Y. Tian, The Moore-Penrose inverses of a triple matrix product, Math. In Theory and Practice 1 (1992), 64–67.
  • Y. Tian, How to characterize equalities for the Moore-Penrose inverses of a matrix, Kyungpook Math. J. 41 (2001), 125–131.
  • G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974), 269–292.
  • P. Patricio, The Moore-Penrose inverse of von Neumann regular matrices over a ring, Linear Algebra Appl. 332 (2001), 469–483.
  • P. Patricio, The Moore-Penrose inverse of a factorization, Linear Algebra Appl. 370 (2003), 227–236.
  • D.W. Robinson, Nullities of submatrices of the Moore-Penrose inverse, Linear Algebra Appl. 94 (1987), 127–132.
  • Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian Matrix, Electron. J. Linear Algebra. 20 (2010), 226–240.
  • L. Zhang, A characterization of the Drazin inverse, Linear Algebra Appl. 335 (2001), 183–188.
  • N. Castro-Gonzalez and E. Dopazo, Representations of the Drazin inverse for a class of block matrices, Linear Algebra Appl. 400 (2005), 253–269.
  • R. E. Harwig, E. Li and Y. Wei, Representations for the Drazin inverse of a block matrix, SIAM J. Matrix Anal. Appl. 27 (2006), 757–771.
  • X. Li and M. Wei, A note on the representations for the Drazin inverse of 2$\times2$ block matrices, Linear Algebra Appl. 423 (2007), 332–338.
  • C. Deng and Y. Wei, New additive results for the generalized Drazin inverse, J. Math. Anal. Appl. 370 (2010), 313–321.
  • S. Dragana and S. Cvetković-Ilić, New additive results on Drazin inverse and its applications, Appl. Math. Comput. 218 (2011), 3019–3024.