Journal of Applied MathematicsVolume 2012 (2012), Article ID 504650, 17 pagesdoi:10.1155/2012/504650

Research Article

The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

Lingling Wu,1,2 Xiaoji Liu,¹ and Yaoming Yu³

¹College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China²College of Mathematics and Computer Science, Bijie University, Guizhou 551700, China³School of Mathematical Sciences, Monash University, Clayton Cambus, VIC 3800, Australia

Received 19 December 2011; Accepted 16 March 2012

Academic Editor: Mehmet Sezer

Copyright © 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., [3–12] 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 and  2.4]). Let be two different nonzero idempotent matrices. Suppose . Then for any scalars , where and , is group invertible, and (i) if , then (ii) if , then

Lemma 1.2 (see [16, Theorem  3.1]). 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)  Under 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)  If , 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)  If , 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)  Consider 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 yieldsIf , 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: (e₁), , ; (e₂), , , .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.

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