International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 2 (2014), 191-197
http://www.etamaths.com

SOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED

MATRIX WITH NEW CONDITIONS

ABDUL SHAKOOR∗, HU YANG AND ILYAS ALI

Abstract. In this article, we consider representations of the Drazin inverse

of a modified matrix M = A − CDdB with the generalized Schur complement
Z = D − BAdC under different conditions given in recent articles on the
subject. Numerical example is given to illustrate our result.

1. Introduction

The importance of the Drazin inverse and its applications to singular differential
equations and difference equations, to Morkov chains and iterative methods, to
cryptography, to numerical analysis, to structured matrices and to perturbation
bounds for the relative eigenvalue problems can be found in [1-3].

Let Cm×n represent the set of m × n complex matrices. Let A ∈ Cn×n, then
there exist a unique matrix Ad ∈ Cn×n satisfying the following equations:

Ak+1Ad = Ak, AdAAd = Ad, AAd = AdA,(1.1)

Ad is called the Drazin inverse of A, where k = ind(A) is the index of A, the smallest
nonnegative integer for which rank(Ak+1) = rank(Ak) (see[1-3]). In particular,
when ind(A) = 1, the Drazin inverse of A is called the group inverse of A. If A is
nonsingular, it is clearly ind(A) = 0 and Ad = A−1. Throughout this article, we
denote by Aπ = I − AAd and define A0 = I, where I is the identity matrix with
proper sizes.

In 1975, Shoaf [4] derived the result of the Drazin inverse of a modified square
matrix, in 1994, Kala et al. [5] gave an explicit representation for the generalized
inverse of a modified matrix, and in 2002, Wei [6] have discussed the expression of
the Drazin inverse of a modified square matrix A − CB. Recently, in 2013, Dopa-
zo et al. [7], Mosić [8] and Shakoor et al. [9] presented some new results for the
Drazin inverse of a modified matrix M = A−CDdB in terms of the Drazin inverse
of the matrix A and the generalized Schur complement Z = D − BAdC under the
following conditions:
(1) AπC = 0, CDπ = 0, DπB = 0, ZπB = 0, CZπ = 0 (see [7]);
(2) BAπ = 0, CDπ = 0, DπB = 0, ZπB = 0, CZπ = 0 (see [7]);
(3) AπC = CDπ, DπB = 0, DZπ = 0 (see [8]);
(4) BAπ = DπB, CDπ = 0, ZπD = 0 (see [8]);
(5) AπC = 0, CDπZdB = 0, CDdZπB = 0, CZdDπB = 0, CZπDdB = 0 (see
[9]);

2010 Mathematics Subject Classification. 15A09, 15A10, 65F20.

Key words and phrases. Drazin inverse; Generalized Schur complement; Modified matrix.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

191



192 SHAKOOR, YANG AND ALI

(6) BAπ = 0, CDπZdB = 0, CDdZπB = 0, CZdDπB = 0 CZπDdB = 0 (see
[9]).
Moreover, Shakoor et al. [9] gave some new results for the Drazin inverse of the
modified matrix M = A − CDdB, when the generalized Schur complement Z = 0
under the following conditions:
(7) AπC = 0, CDπΓdB = 0, CDdΓπB = 0, CΓdDπB = 0 CΓπDdB = 0 (see
[9]);
(8) BAπ = 0, CDπΓdB = 0, CDdΓπB = 0, CΓdDπB = 0 CΓπDdB = 0 (see [9]).

In this article, we consider the Drazin inverse of a modified matrix M = A −
CDdB in terms of the Drazin inverse of the matrix A and the generalized Schur
complement Z = D−BAdC under conditions weaker than conditions (5) and (6) in
[9], which extends some results in [7,8]. Furthermore, we consider some results for
the Drazin inverse of the modified matrix M = A − CDdB, when the generalized
Schur complement Z = 0 under different conditions in [9]. Finally, we give a
numerical example to illustrate our result.

2. The Drazin inverse of a modified matrix

In this section, we consider the Drazin inverse of a modified matrix M = A −
CDdB in terms of the Drazin inverse of the matrix A and the generalized Schur
complement Z = D−BAdC is not necessarily invertible under different conditions
presented in [7,8,9].

Let A, B, C, D ∈ Cn×n. Throughout this section, we use the following notations:
M = A−CDdB, Z = D −BAdC(2.1)

and

K = AdC, H = BAd, Γ = HK.(2.2)

First, we present the following theorem.

Theorem 2.1. Let A, B, C, and D be complex matrices, where ind(A) = k. If
AπC = CDπ, CDdZπB = 0, CZdDπB = 0 and CZπDdB = 0, then

Md = Ad + KZdH −
k−1∑
i=0

(Ad + KZdH)i+1KZdBAiAπ(2.3)

and ind(M) ≤ ind(A).

Proof. Let X = Ad + KZdH. The assumption AπC = CDπ implies that AAdC =
CDDd. Firstly, we note the facts:

MX = AAd + AAdCZdBAd −CDdBAd −CDd(D −Z)ZdBAd

= AAd + CDDdZdBAd −CDdDZdBAd −CDdZπBAd

= AAd(2.4)

and

XM = AdA + AdCZdBAdA−AdCDdB −AdCZd(D −Z)DdB
= AdA−AdCZdBAπ −AdCZπDdB
= AdA−KZdBAπ.(2.5)



ON THE DRAZIN INVERSE OF A MODIFIED MATRIX 193

From (2.4), we have

MMd = MX −MX
k−1∑
i=0

XiKZdBAiAπ

= AAd −
k−1∑
i=0

XiKZdBAiAπ(2.6)

and using (2.5), we get

MdM = XM −
k−1∑
i=0

Xi+1KZdBAiAπM

= AdA−KZdBAπ −
k−1∑
i=0

Xi+1KZdBAi+1Aπ

= AdA−
k−1∑
i=0

XiKZdBAiAπ.(2.7)

Thus

MMd = MdM.

Secondly, from (2.7) and AπMd = 0, we obtain

MdMMd = (AdA−
k−1∑
i=0

XiKZdBAiAπ)Md

= AdAMd

= Md.

Finally, we shall prove that M −M2Md is a nilpotent matrix. From (2.6), we get

M −M2Md = [I −MMd]M

= (I −AAd +
k−1∑
i=0

XiKZdBAiAπ)M

= AAπ +

k−1∑
i=0

XiKZdBAi+1Aπ.

By induction on integer n ≥ 1, we have

(M −M2Md)n = AnAπ +
k−1∑
i=0

XiKZdBAi+nAπ.(2.8)

From (2.8), it gives that (M − M2Md)k = 0, where k = ind(A). Therefore, we
conclude that Mk+1Md = Mk and ind(M) ≤ ind(A), that completes the proof. �

From Theorem 2.1, we obtain the following corollaries.



194 SHAKOOR, YANG AND ALI

Corollary 2.2 ([8]). Let A, B, C, and D be complex matrices, where ind(A) = k.
If AπC = CDπ, ZπB = 0, DπB = 0 and CZπ = 0, then

Md = Ad + KZdH −
k−1∑
i=0

(Ad + KZdH)i+1KZdBAiAπ

and ind(M) ≤ ind(A).

Corollary 2.3 ([8]). Let A, B, C, and D be complex matrices, where ind(A) = k.
If AπC = CDπ, DZπ = 0 and DπB = 0, then

Md = Ad + KZdH −
k−1∑
i=0

(Ad + KZdH)i+1KZdBAiAπ

and ind(M) ≤ ind(A).

In the same way, we give a new theorem.

Theorem 2.4. Let A, B, C, and D be complex matrices, where ind(A) = k. If
BAπ = DπB, CDπZdB = 0, CDdZπB = 0 and CZπDdB = 0, then

Md = Ad + KZdH −
k−1∑
i=0

AπAiCZdH(Ad + KZdH)i+1

and ind(M) ≤ ind(A).

Similarly, from Theorem 2.2. we have the following corollaries.

Corollary 2.5 ([8]). Let A, B, C, and D be complex matrices, where ind(A) = k.
If BAπ = DπB, CDπ = 0, ZπB = 0 and CZπ = 0, then

Md = Ad + KZdH −
k−1∑
i=0

AπAiCZdH(Ad + KZdH)i+1

and ind(M) ≤ ind(A).

Corollary 2.6 ([8]). Let A, B, C, and D be complex matrices, where ind(A) = k.
If BAπ = DπB, CDπ = 0 and ZπD = 0, then

Md = Ad + KZdH −
k−1∑
i=0

AπAiCZdH(Ad + KZdH)i+1

and ind(M) ≤ ind(A).

Now, we consider some results for the Drazin inverse of the modified matrix
M = A − CDdB, when the generalized Schur complement Z = 0 under different
conditions in [9].



ON THE DRAZIN INVERSE OF A MODIFIED MATRIX 195

Theorem 2.7. Let A, B, C, and D be complex matrices, where ind(A) = k. If
Z = 0, AπC = CDπ, DπB = 0 and DΓπ = 0, then

Md = (I −KΓdH)Ad(I −KΓdH) +
k−1∑
i=0

[(I −KΓdH)Ad]i+2KΓdBAiAπ

(2.9)

and ind(M) ≤ ind(A).

Proof. Let X = (I − KΓdH)Ad(I − KΓdH). The assumptions AπC = CDπ and
DΓπ = 0 imply that AAdC = CDDd and DdΓπ = 0. Firstly, we note the facts:

MX = (A−AAdCΓdBAd −CDdB + CDdDΓdBAd)Ad(I −KΓdH)
= (A−CDdB)(Ad − (Ad)2CΓdBAd)
= AAd −AdCΓdBAd −CDdBAd + CDdΓΓdBAd

= AAd −KΓdH −CDdΓπBAd

= AAd −KΓdH.(2.10)
Since DDdB = B, AdC = AdCDDd and DΓπ = 0, then

XM = (I −KΓdH)Ad(A−CDdB −AdCΓdBAdA + AdCΓdDDdB)
= (I −KΓdH)Ad(A−CDdB −AdCΓdBAdA + AdCΓdB)
= (Ad −AdCΓdB(Ad)2)(A−CDdB + AdCΓdBAπ)
= AdA−AdCDdB + (Ad)2CΓdBAπ −AdCΓdBAd + AdCΓdΓDdB
−AdCΓdB(Ad)3CΓdBAπ

= AdA−KΓdH + (I −AdCΓdBAd)(Ad)2CΓdBAπ −AdCDdDΓπDdB
= AdA−KΓdH + (I −KΓdH)AdKΓdBAπ.(2.11)

From (2.10), we have

MMd = MX + M(I −KΓdH)Ad
k−1∑
i=0

[(I −KΓdH)Ad]i+1KΓdBAiAπ

= AAd −KΓdH + (AAd −CDdBAd)
k−1∑
i=0

[(I −KΓdH)Ad]i+1KΓdBAiAπ

= AAd −KΓdH +
k−1∑
i=0

[(I −KΓdH)Ad]i+1KΓdBAiAπ(2.12)

and using (2.11), we get

MdM = XM +

k−1∑
i=0

[(I −KΓdH)Ad]i+2KΓdBAiAπM

= AdA−KΓdH + (I −KΓdH)AdKΓdBAπ +
k−1∑
i=0

[(I −KΓdH)Ad]i+2

×KΓdBAi+1Aπ

= AdA−KΓdH +
k−1∑
i=0

[(I −KΓdH)Ad]i+1KΓdBAiAπ.(2.13)



196 SHAKOOR, YANG AND ALI

Thus

MMd = MdM.

From (2.13) and AπMd = 0, we obtain

MdMMd = (AdA−KΓdH +
k−1∑
i=0

[(I −KΓdH)Ad]i+1KΓdBAiAπ)Md

= (AdA−KΓdH)Md

= Md.

By using (2.12) and DπB = 0, we get

M −M2Md = AAπ −
k−1∑
i=0

[(I −KΓdH)Ad]iKΓdBAi+1Aπ.

By induction on integer n ≥ 1, we have

(M −M2Md)n = AnAπ −
k−1∑
i=0

[(I −KΓdH)Ad]iKΓdBAi+nAπ.

From above expression, it follows that (M − M2Md)k = 0, where k = ind(A).
Therefore, we obtain that Mk+1Md = Mk and ind(M) ≤ ind(A), which completes
the proof. �

In similar way, we present another result of this paper.

Theorem 2.8. Let A, B, C, and D be complex matrices, where ind(A) = k. If
Z = 0, BAπ = DπB, CDπ = 0 and ΓπD = 0, then

Md = (I −KΓdH)Ad(I −KΓdH) +
k−1∑
i=0

AπAiCΓdH[Ad(I −KΓdH)]i+2

and ind(M) ≤ ind(A).

In the end of this section, we give a numerical example to demonstrate our re-
sult of a modified matrix. This numerical example describes matrices A, B, C and
D which do not satisfy the conditions of [7, Theorem 2.1] nor the conditions of
[8, Theorem 1] but they satisfy the conditions of Theorem 2.1. Therefore, we can
apply the formula given in Theorem 2.1 to obtain the Drazin inverse of a modified
matrix M.

Numerical example 2.9. Consider the matrices

A =


 1 0 00 0 0

0 0 0


, B = ( 2 1 1

0 0 0

)
, C =


 −1 −10 0

0 0


, D = ( 1 1

0 0

)
.

Note that ind(A) = 1 and ind(D) = 1, then we obtain

Ad =


 1 0 00 0 0

0 0 0


 , Aπ =


 0 0 00 1 0

0 0 1


 , Dd = ( 1 1

0 0

)
, Dπ =

(
0 −1
0 1

)
.



ON THE DRAZIN INVERSE OF A MODIFIED MATRIX 197

Now we have

M = A−CDdB =


 3 1 10 0 0

0 0 0




and

Z = D −BAdC =
(

3 1
0 0

)
, Zd =

1

9

(
3 1
0 0

)
, Zπ =

1

3

(
0 −1
0 3

)
.

It can be calculated that:
(i) CZπ 6= 0, so the conditions given in [7, Theorem 2.1] are not satisfied.
(ii) DZπ 6= 0, thus the conditions given in [8, Theorem 1] are not satisfied.

On the other hand, we can observe that AπC = CDπ, CDdZπB = 0, CZdDπB =
0 and CZπDdB = 0. Then applying Theorem 2.1, we obtain

Md =
1

9


 3 1 10 0 0

0 0 0


 .

3. Acknowledgment

This work was supported by the Ph.D. programs Foundation of Ministry of
Education of China (Grant No. 20110191110033).

References

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Springer, New York, 2003.
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1991.

[3] G. Wang, Y. Wei, S. Qiao, Generalized Inverses: Theory and Computations, Science Press,
Beijing/New York, 2004.

[4] J.M. Shoaf, The Drazin inverse of a rank-one modification of a square matrix, Ph.D. Disser-

tation, North Carolina State University, 1975.
[5] R. Kala, K. Klaczyński, Generalized inverses of a sum of matrices, Sankhya Ser. A 56 (1994)

458-464.

[6] Y. Wei, The Drazin inverse of a modified matrix, Appl. Math. Comput. 125 (2002) 295-301.
[7] E. Dopazo, M.F. Mart́ınez-Serrano, On deriving the Drazin inverse of a modified matrix,

Linear Algebra Appl. 438 (2013) 1678-1687.
[8] D. Mosić, Some results on the Drazin inverse of a modified matrix, Calcolo. 50 (2013) 305?11.

[9] A. Shakoor, H. Yang, I. Ali, Some representations for the Drazin inverse of a modified matrix,

Calcolo. DOI 10.1007/s10092-013-0098-0 (2013).

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331,
China

∗Corresponding author