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

We give a formula of(P+Q)Dunder the conditionsP2Q+QPQ=0,P3Q=0, andPQPQ=0. Then, we apply it to give some expressions for the Drazin inverse of block matrixM=(ABCD)(AandDare square matrices) under some conditions, generalizing some recent results in the literature. Finally, numerical examples are given to illustrate our results.

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On some previous results for the Drazin inverse of block matrices

Filomat ◽

10.2298/fil1601125v ◽

2016 ◽

Vol 30 (1) ◽

pp. 125-130 ◽

Cited By ~ 1

Author(s):

Jelena Visnjic

Keyword(s):

Block Matrix ◽

Drazin Inverse ◽

Short Paper ◽

Block Matrices ◽

Matrix Anal ◽

Appl Math

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

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A note on the representation for the Drazin inverse of 2×2 block matrices

Linear Algebra and its Applications ◽

10.1016/j.laa.2008.02.019 ◽

2008 ◽

Vol 429 (1) ◽

pp. 242-248 ◽

Cited By ~ 39

Author(s):

Dragana S. Cvetković-Ilić

Keyword(s):

Drazin Inverse ◽

Block Matrices

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The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498819502128 ◽

2019 ◽

Vol 18 (11) ◽

pp. 1950212 ◽

Cited By ~ 1

Author(s):

Honglin Zou ◽

Dijana Mosić ◽

Jianlong Chen

Keyword(s):

Sufficient Conditions ◽

General Setting ◽

Upper Bounds ◽

Block Matrix ◽

Drazin Inverse ◽

Operator Matrices ◽

Skew Field ◽

Block Matrices ◽

Necessary And Sufficient ◽

Drazin Index

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

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