Missouri Journal of Mathematical Sciences

The Nullity and Rank of Combinations of Two Outer Inverses of a Given Matrix

Kezheng Zuo and Tao Xie

Full-text: Open access

Abstract

This paper shows that the nullity and rank of $aP+bQ-cQAP$ is a constant, where $P$ and $Q$ are outer inverses of a given matrix $A$, $c=a+b$ ($a,b\neq 0$) or $c\neq a+b$, $a, b, c \in \mathbb{C}$. In addition, the rank of $aP+bQ-cQAP$ is equal to the rank of $P-Q$ if $c=a+b$ and to $P+Q$ if $c\neq a+b$.

Article information

Source
Missouri J. Math. Sci., Volume 22, Issue 1 (2010), 19-22.

Dates
First available in Project Euclid: 1 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1312232717

Digital Object Identifier
doi:10.35834/mjms/1312232717

Mathematical Reviews number (MathSciNet)
MR2650058

Zentralblatt MATH identifier
1202.15007

Subjects
Primary: 15A03: Vector spaces, linear dependence, rank
Secondary: 15A03: Vector spaces, linear dependence, rank

Citation

Zuo, Kezheng; Xie, Tao. The Nullity and Rank of Combinations of Two Outer Inverses of a Given Matrix. Missouri J. Math. Sci. 22 (2010), no. 1, 19--22. doi:10.35834/mjms/1312232717. https://projecteuclid.org/euclid.mjms/1312232717


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References

  • A. Ben-Isrel and T. N. G. Greville, Generalized inverses: theory and applications, Springer, 2003, second edition.
  • Y. Chen and X. Chen, Representation and approximation of the outer inverse $A_{T,S}^{(2)}$ of a matrix $A$, Linear Algebra Appl., 308 (2000), 85–107.
  • E. P. Liski and S. Wang, On the \2\-inverse and some ordering properties of nonnegative definite matrices, Acta Math. Appl. Sinica (English Series), 12 (1996), 22–27.
  • J. J. Koliha and V. Rakočević, The nullity and rank of linear combinations of idempotent matrices, Linear Algebra Appl., 418 (2006), 11–14.
  • J. Gro$\beta$ and G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl., 21 (1999), 390–395.
  • Y. Tian and G. P. H. Styan, Rank equalities for idempotent and involutary matrices, Linear Algebra Appl., 335 (2001), 101–117.
  • Y. Tian, Rank equalities related to outer inverses of matrices and applications, Linear Algebra Appl., 388 (2004), 279–288.
  • Y. Tian, More on rank equalities for outer inverses of matrices with applications, International Journal of Mathematics, Game Theory and Algebra, 12 (2002), 137–151.
  • Zuo Kezhing, The nullity and rank of combinations of idempotent matrices, Journal of Math., 28 (2008), 619–622.