Some additive results for the Drazin inverse and its application

In this paper we derive the formula of (P + Q)D under the conditions Q(P + Q)P(P + Q) = 0, P(P + Q)P(P + Q) = 0 and QPQ2 = 0. Then, a corollary is given which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Meanwhile, we show that the additive formula provided by Bu et al. (J. Appl. Math. Comput. 38 (2012) 631-640) is not valid for all matrices which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Also, the representation can be simplified from Višnjić (Filomat 30 (2016) 125-130) which satisfies given conditions. Furthermore, we apply our result to establish a new representation for the Drazin inverse of a complex block matrix having generalized Schur complement equal to zero under some conditions. Finally, a numerical example is given to illustrate our result.

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On a recent result concerning the Drazin inverse of matrices

Višnjić

Ilić

2022

Filomat

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In this paper, we investigate the recent paper of Shakoor et al. [A. Shakoor, I. Ali, S. Wali, A. Rehman, Some formulas on the Drazin inverse for the sum of two matrices and block matrices, Bull. Iran. Math. Soc. 48 (2022) 351-366]. Here we prove that the main, additive result from the mentioned paper is actually a corollary of one known result. Furthermore, we give new representations for the Drazin inverse of anti-triangular block matrix, which generalize some representations from current literature on the topic.

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Some additive results for the Drazin inverse and its application

Abstract: In this paper, we give some formulas for the Drazin inverses of the sum of two matrices under conditions weaker than those used in some current literature. Also, we obtain representation for the Drazin inverse of a complex block matrix under some conditions.

Cited by 4 publications

References 5 publications

Some Formulas on the Drazin Inverse for the Sum of Two Matrices and Block Matrices

Some Formulas on the Drazin Inverse for the Sum of Two Matrices and Block Matrices

A note on the formulas for the Drazin inverse of the sum of two matrices

On a recent result concerning the Drazin inverse of matrices

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