How to characterize commutativity equalities for Drazin inverses of matrices

Yong Ge Tian

Archivum Mathematicum (2003)

  • Volume: 039, Issue: 3, page 191-199
  • ISSN: 0044-8753

Abstract

top
Necessary and sufficient conditions are presented for the commutativity equalities A * A D = A D A * , A † A D = A D A † , A † A A D = A D A A † , A A D A * = A * A D A and so on to hold by using rank equalities of matrices. Some related topics are also examined.

How to cite

top

Tian, Yong Ge. "How to characterize commutativity equalities for Drazin inverses of matrices." Archivum Mathematicum 039.3 (2003): 191-199. <http://eudml.org/doc/249133>.

@article{Tian2003,
abstract = {Necessary and sufficient conditions are presented for the commutativity equalities $A^*A^D = A^DA^*$, $A^\{\dag \}A^D = A^DA^\{\dag \}$, $A^\{\dag \}AA^D = A^DAA^\{\dag \}$, $AA^DA^* = A^*A^DA$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.},
author = {Tian, Yong Ge},
journal = {Archivum Mathematicum},
keywords = {commutativity; Drazin inverse; Moore-Penrose inverse; rank equality; matrix expression; commutativity; Drazin inverse; Moore-Penrose inverse; rank equality; matrix expression},
language = {eng},
number = {3},
pages = {191-199},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {How to characterize commutativity equalities for Drazin inverses of matrices},
url = {http://eudml.org/doc/249133},
volume = {039},
year = {2003},
}

TY - JOUR
AU - Tian, Yong Ge
TI - How to characterize commutativity equalities for Drazin inverses of matrices
JO - Archivum Mathematicum
PY - 2003
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 039
IS - 3
SP - 191
EP - 199
AB - Necessary and sufficient conditions are presented for the commutativity equalities $A^*A^D = A^DA^*$, $A^{\dag }A^D = A^DA^{\dag }$, $A^{\dag }AA^D = A^DAA^{\dag }$, $AA^DA^* = A^*A^DA$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.
LA - eng
KW - commutativity; Drazin inverse; Moore-Penrose inverse; rank equality; matrix expression; commutativity; Drazin inverse; Moore-Penrose inverse; rank equality; matrix expression
UR - http://eudml.org/doc/249133
ER -

References

top
  1. Ben-Israel A., Greville T. N. E., Generalized Inverses: Theory and Applications, Corrected reprint of the 1974 original, Robert E. Krieger Publishing Co., Inc., Huntington, New York, 1980. (1974) Zbl0305.15001MR0587113
  2. Campbell S. L., Meyer C. D., EP operators and generalized inverses, Canad. Math. Bull. 18 (1975), 327–333. (1975) Zbl0317.15004MR0405136
  3. Campbell S. L., Meyer C. D., Generalized Inverses of Linear Transformations, Corrected reprint of the 1979 original, Dover Publications, Inc., New York, 1991. (1979) Zbl0417.15002MR1105324
  4. Castro N., Koliha J. J., Perturbation of the Drazin inverse for closed linear operators, Integral Equations Operator Theory 36 (2000), 92–106. Zbl1009.47002MR1736919
  5. Castro N., Koliha J. J., Wei Y., Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero, Linear Algebra Appl. 312 (2000), 181–189. Zbl0963.15002MR1759331
  6. Hartwig R. E., Spindelböck K., Partial isometries, contractions and EP matrices, Linear and Multilinear Algebra 13 (1983), 295–310. (1983) Zbl0575.15008MR0704779
  7. Hartwig R. E., Spindelböck K., Matrices for which A * and A † can commute, Linear and Multilinear Algebra 14 (1984), 241–256. (1984) 
  8. Koliha J. J., Elements of C * -algebras commuting with their Moore-Penrose inverse, Studia Math. 139 (2000), 81–90. Zbl0963.46037MR1763046
  9. Marsaglia G., Styan G. P. H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2 (1974), 269–292. (1974) Zbl0297.15003MR0384840
  10. Rao C. R., Mitra S. K., Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971. (1971) Zbl0236.15005MR0338013
  11. Tian Y., How to characterize equalities for the Moore-Penrose inverse of a matrix, Kyungpook Math. J. 41 (2001), 1–15. Zbl0987.15001MR1847431
  12. Wong E. T., Does the generalized inverse of A commute with A ?, Math. Mag. 59 (1986), 230–232. (1986) Zbl0611.15007MR1572626

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.