Journal of Applied Mathematics

VolumeΒ 2012, Article IDΒ 504650, 17 pages

http://dx.doi.org/10.1155/2012/504650

## The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

^{1}College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China^{2}College of Mathematics and Computer Science, Bijie University, Guizhou 551700, China^{3}School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800, Australia

Received 19 December 2011; Accepted 16 March 2012

Academic Editor: MehmetΒ Sezer

Copyright Β© 2012 Lingling Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We discuss the following problem: when of idempotent matrices and , where and , is group involutory.

#### 1. Introduction

Throughout this paper stands for the set of complex matrices. Let . is said to be idempotent if . is said to be group invertible if there exists an such that hold. If such an exists, then it is unique, denoted by , and called the group inverse of . It is well known that the group inverse of a square matrix exists if and only if (see, e.g., [1] for details). Clearly, not every matrix is group invertible. But the group inverse of every idempotent matrix exists and is this matrix itself.

Recall that a matrix with the group inverse is said to be group involutory if . is the group involutory matrix if and only if it is tripotent, that is, satisfies (see [2]). Thus, for a nonzero idempotent matrix and a nonzero scalar , is a group involutory matrix if and only if either or .

Recently, some properties of linear combinations of idempotents or projections are widely discussed (see, e.g., [3β12] and the literature mentioned below). In [13], authors established a complete solution to the problem of when a linear combination of two different projectors is also a projector. In [14], authors considered the following problem: when a linear combination of nonzero different idempotent matrices is the group involutory matrix. In [15], authors provided the complete list of situations in which a linear combination of two idempotent matrices is the group involutory matrix. In [16], authors discussed the group inverse of of idempotent matrices and , where with , deduced its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of .

In this paper, we will investigate the following problem: when is group involutory. To this end, we need the results below.

Lemma 1.1 (see [16, Theorems 2.1 andββ2.4]). *Let be two different nonzero idempotent matrices. Suppose . Then for any scalars , where and , is group invertible, and**
(i)βif , then
**
(ii) if , then
*

Lemma 1.2 (see [16, Theoremββ3.1]). *Let be two different nonzero idempotent matrices. Suppose . Then for any scalars , and , where , is group invertible, and
*

#### 2. Main Results

In this section, we will research when some combination of two nonzero idempotent matrices is a group involutory matrix.

First, we will discuss some situations lying in the category of .

Theorem 2.1. *Let be two different nonzero idempotent matrices with , and let be a combination of the form
**
where with . Denote . Then the following list comprises characteristics of all cases where is the group involutory matrix:**(a) the cases denoted by , in which
**
and any of the following sets of additional conditions hold:*(*a*_{1})*either or , either or or , and ;*(*a*_{2})*either or , either or or , and ;*(*a*_{3})*either or , either or , either or or or .**(b) the cases denoted by , in which
**
and any of the following sets of additional conditions hold:*(*b*_{1})*, either or or or ; *(*b*_{2})*, either or or or ;*(*b*_{3})*, either or or , and ;*(*b*_{4})*, either or or , and ;*(*b*_{5})*, , and ;*(*b*_{6})*, , and ,**(c) the cases denoted by , in which
**
and any of the following sets of additional conditions hold:*(*c*_{1})*, either or , and ; *(*c*_{2})*, either or , and ;*(*c*_{3})*, either or , and ;*(*c*_{4})*, either or , and ;*(*c*_{5})*, either or ;*(*c*_{6})*, either or ;*(*c*_{7})*, and ;*(*c*_{8})*, and ;*(*c*_{9})*, and ;*(*c*_{10})*, and ;*(*c*_{11})*, and ;*(*c*_{12})*, and ;*(*c*_{13})*, and ;*(*c*_{14})*, and ;*(*c*_{15})*, and ;*(*c*_{16})*, and ;*(*c*_{17})*;*(*c*_{18})*.*

* Proof. *Obviously, the condition (2.2) implies that the group inverse of exists and is of the form (1.2) when or the form (1.3) when by Lemma 1.1. So do the conditions (2.2), (2.3), and (2.4). We will straightforwardly show that a matrix of the form (2.1) is the group involutory matrix if and only if .

(a)ββUnder the condition (2.2), , where .

(1) If , then
and so
Multiplying (2.6) by and , respectively, leads to
and then
Multiplying the above equation, respectively, by and by , we get
Thus, since , we have three situations: and ; and ; and .

When and , (2.6) becomes and then . Therefore, we obtain except the situation . Similarly, when and , we have except the situation . When and , (2.6) becomes and then or . Therefore, we obtain except the situation .

(2) If , then
and then
Analogous to the process of reaching (2.9) in , we have
Thus, we have three situations: and ; and ; and , since . Similar to the argument in , substituting them, respectively, into (2.11), we can obtain the situation , respectively, in , and .

(b) Under the condition (2.3), , where .

(1) If , then
and so
Multiplying the above equation, respectively, on the two sides by yields
Multiplying (2.15) on the left sides by and (2.16) on the right sides by , by (2.3), we have
and then . Since , . Similarly, .

Substituting inside (2.17) yields and then or . We will discuss the remainder for detail as follows:

When , , (2.14) becomes

(i) if , then
and so it follows from (2.18) that
Therefore, either or implies that (2.18) holds, namely, (2.14) holds. Thus, we have except the situation .

(ii) if , then (2.18) becomes
Multiplying the above equation, respectively, on the right side by and on the left side by , we have
So if , then the two equations above (2.22) and (2.23) become, respectively,
Or if , then (2.22) and (2.23) become, respectively,
Since , it follows from (2.24) and (2.25) that we have the six situations: and ; , and ; , , and ; and ; and ; and . Thus, we have except the situation , and and .

(2) If , then
and then
Analogous to the process in , using (2.27) we can obtain
Thus, since , and/or and then . Similarly, . Hence, .

(i)ββIf , then
Thus, (2.27) holds. Hence we have the situation in .

(ii) If , then (2.27) becomes
Multiplying the above equation on the left side, respectively, by and by , we have
Thus, ; and ; and . Hence, we have the situation , respectively, in , and .

(c) Under the condition (2.4),

(1)ββIf , then
and so
If , then and so it contradicts (2.4). Thus . Similarly, .

Multiplying (2.34) on the left side by yields
Multiplying the above equation, respectively, on the left side by and on the right side by yields, by (2.4),
Since , by (2.36) and (2.37). Similarly, we can gain . Substituting inside (2.36) yields or .

(i)ββConsider the case of , and .

Substituting , , and inside (2.35) yields
Similarly, we have

If , then by the hypothesis and so by (2.38). Multiplying (2.34) on the right side by yields
Thus, and then (2.14) becomes
Multiplying the above equation on the right side by yields
Assume . Then and it contradicts the hypothesis . Thus, .

Similarly, if , then we can obtain , , and .

Obviously, if and , we have , , , and .

Next, we calculate these scalars. If , then for any and for any , and so are chosen to satisfy . Similarly are chosen to satisfy .

If , then , and by solving , and by solving .

Note that and imply . Hence, we have .

(ii) Consider the case of , , and .

Multiplying (2.34), respectively, on the right side by and on the left side by yields

If , then and so and (2.34) becomes
Multiplying (2.44) on right side by yields
Since , and then (2.44) becomes
Thus, .

If , then we, similarly, have , , and .

If and , then, multiplying (2.34), on the right side by and on the left side by yields , and multiplying (2.34) on the right side by and on the left side by yields . Thus, (2.34) becomes
Multiplying the equation above on the right side, respectively, by and by yields
As the argument above in (i), we have .

(2) If , then
and so
Analogous to the process in (c)(1), using (2.50), we can get

Thus, since , and/or and then . Similarly, . Therefore, multiplying (2.50) on the right side by and on the left side by yields
Multiplying (2.50) on the right side by and on the left side by yields
Since and , and . Multiplying (2.50) on the left side, respectively, by and by yields
Thus, we have and ; and ; and .

Note that and imply by . As the argument above in , we have .

*Remark 2.2. *Clearly, [15, (a) and (b) in Theorem] are the special cases in Theorem 2.1.

*Example 2.3. *Let
Then they, obviously, are idempotent, and but . By Theorem 2.1,
is the group involutory matrix, namely, , since and . By Theorem 2.1,
is group involutory since and .

Next, we will study the situation or .

Theorem 2.4. *Let be two different nonzero idempotent matrices, and let be a combination of the form
**
where with . Suppose that
**
and any of the following sets of additional conditions hold:*

, , ;

, , , .

Then is the group involutory matrix.

*Proof. *By Lemma 1.2,
Since , multiplying (2.60), respectively, on the right side and on the right side by yields
and so and . Substituting them inside (2.60), we get
Multiplying (2.62) on the left side by yields
and then
So (2.62) becomes
Multiplying (2.65) on the left side by and on the right side by yields
Therefore,
Similarly, we can obtain
By (2.64) and (2.67), we can obtain
Since and , . If , then (2.64) holds for any , (2.67) holds for any , and, for any satisfying (2.69) and any ,
If , then, by (2.64) *~* (2.69), and and so from (2.68).

Hence, we have and .

*Example 2.5. *Let
Obviously they are idempotent, and but . By Theorem 2.4,
is group involutory since and .

Similarly, we have the following result.

Theorem 2.6. *Let be two different nonzero idempotent matrices, and let be a combination of the form
**
where with . Suppose that
**
and any of the following sets of additional conditions hold:**(e _{1})*

*, , ;*

*(e*

_{2})*, , , .*

*Then is the group involutory matrix.*

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China (11061005) and the Ministry of Education Science anf Technology Key Project (210164) and Grants (HCIC201103) of Guangxi Key Laborarory of Hybrid Computational and IC Design Analysis Open Fund.

#### References

- A. Ben-Israel and T. N. E. Greville,
*Generalized Inverses: Theory and Applications*, Springer, New York, NY, USA, 2nd edition, 2003. - R. Bru and N. Thome, βGroup inverse and group involutory matrices,β
*Linear and Multilinear Algebra*, vol. 45, no. 2-3, pp. 207β218, 1998. View at Publisher Β· View at Google Scholar - J. K. Baksalary and O. M. Baksalary, βOn linear combinations of generalized projectors,β
*Linear Algebra and its Applications*, vol. 388, pp. 17β24, 2004. View at Publisher Β· View at Google Scholar - J. K. Baksalary, O. M. Baksalary, and H. Γzdemir, βA note on linear combinations of commuting tripotent matrices,β
*Linear Algebra and its Applications*, vol. 388, pp. 45β51, 2004. View at Publisher Β· View at Google Scholar - J. K. Baksalary, O. M. Baksalary, and G. P. H. Styan, βIdempotency of linear combinations of an idempotent matrix and a tripotent matrix,β
*Linear Algebra and its Applications*, vol. 354, pp. 21β34, 2002, Ninth special issue on linear algebra and statistics. View at Publisher Β· View at Google Scholar - O. M. Baksalary and J. BenΓtez, βIdempotency of linear combinations of three idempotent matrices, two of which are commuting,β
*Linear Algebra and its Applications*, vol. 424, no. 1, pp. 320β337, 2007. View at Publisher Β· View at Google Scholar - J. BenΓtez and N. Thome, βIdempotency of linear combinations of an idempotent matrix and a
*t*-potent matrix that commute,β*Linear Algebra and its Applications*, vol. 403, pp. 414β418, 2005. View at Publisher Β· View at Google Scholar - J. BenΓtez and N. Thome, βIdempotency of linear combinations of an idempotent matrix and a
*t*-potent matrix that do not commute,β*Linear and Multilinear Algebra*, vol. 56, no. 6, pp. 679β687, 2008. View at Publisher Β· View at Google Scholar - Y. N. Chen and H. K. Du, βIdempotency of linear combinations of two idempotent operators,β
*Acta Mathematica Sinica*, vol. 50, no. 5, pp. 1171β1176, 2007. View at Google Scholar - J. J. Koliha, V. RakoΔeviΔ, and I. StraΕ‘kraba, βThe difference and sum of projectors,β
*Linear Algebra and its Applications*, vol. 388, pp. 279β288, 2004. View at Publisher Β· View at Google Scholar - H. Γzdemir and A. Y. Γzban, βOn idempotency of linear combinations of idempotent matrices,β
*Applied Mathematics and Computation*, vol. 159, no. 2, pp. 439β448, 2004. View at Publisher Β· View at Google Scholar - M. Sarduvan and H. Γzdemir, βOn linear combinations of two tripotent, idempotent, and involutive matrices,β
*Applied Mathematics and Computation*, vol. 200, no. 1, pp. 401β406, 2008. View at Publisher Β· View at Google Scholar - J. K. Baksalary and O. M. Baksalary, βIdempotency of linear combinations of two idempotent matrices,β
*Linear Algebra and its Applications*, vol. 321, no. 1β3, pp. 3β7, 2000, Linear algebra and statistics (Fort Lauderdale, FL, 1998). View at Publisher Β· View at Google Scholar - C. Coll and N. Thome, βOblique projectors and group involutory matrices,β
*Applied Mathematics and Computation*, vol. 140, no. 2-3, pp. 517β522, 2003. View at Publisher Β· View at Google Scholar - J. K. Baksalary and O. M. Baksalary, βWhen is a linear combination of two idempotent matrices the group involutory matrix?β
*Linear and Multilinear Algebra*, vol. 54, no. 6, pp. 429β435, 2006. View at Publisher Β· View at Google Scholar - X. Liu, L. Wu, and Y. Yu, βThe group inverse of the combinations of two idempotent matrices,β
*Linear and Multilinear Algebra*, vol. 59, no. 1, pp. 101β115, 2011. View at Publisher Β· View at Google Scholar