#### Abstract

Tian and Styan have shown many rank equalities for the sum of two and three idempotent matrices and pointed out that rank equalities for the sum with be idempotent () are still open. In this paper, by using block Gaussian elimination, we obtained rank equalities for the sum of finitely many idempotent matrices and then solved the open problem mentioned above. Extensions to scalar-potent matrices and some related matrices are also included.

#### 1. Introduction

Let and be the sets of complex matrices and nonsingular matrices, respectively. The identity matrix is denoted by or simply by if the size is immaterial. Let be the set of all the positive integer numbers. The symbols and stand for the rank and transpose of , respectively, while denotes the trace of a square matrix . A matrix is said to be* idempotent*, if , and* scalar*-*potent* (determined by ), if , for some (see, e.g., [1]). When , it coincides with the definition of an idempotent matrix.

As one of the fundamental building blocks in matrix theory, idempotent matrices are very useful in many contexts and have been extensively studied in the literature (see, e.g., [1–6]). Here we focus on the research on the rank of the sum of idempotent matrices.

Gröss and Trenkler have studied rank of the sum of two idempotent matrices (see [3, Theorem 3]). Also, Tian and Styan have shown a rank equality for two idempotent matrices as follows.

Proposition 1 (see [1, Theorem 2.4] and [2, Theorem 2.1]). *Let be idempotent. Then
*

Tian and Styan have extended the rank equality for the sum of idempotent matrices to the scalar-potent matrices (see, e.g., [1]).

Proposition 2 (see [1, P110]). *Let be scalar-potent matrices determined by nonzero complexes . Then
*

Later, Tian and Styan considered the rank equality for the sum of three idempotent matrices in [2] as follows.

Proposition 3 (see [2, P95]). *Let be idempotent. Then
*

By (3), Tian and Styan have induced many useful results, for example, if , , are idempotent and , then . The literatures [2, 4–6] show that establishing various kinds of rank equalities for idempotent matrices is interesting. Tian and Styan pointed out that rank equalities for the sum with be idempotent () are still open (see [2, P95]).

In this paper, by applying block Gaussian elimination, rank equalities for the sum of finitely many idempotent matrices are obtained. These results generalize (3) and solve the open problem proposed by Tian and Styan (see, e.g., [2]). Also, new rank equalities for finitely many idempotent matrices are given. The rank equality (3) is generalized to scalar-potent matrices as well.

#### 2. Main Results

Before showing main results, we need some preparations.

Lemma 4. *Let . Then
**
for any .*

*Proof. *Let
It is evident that and are nonsingular.

By calculation,
since and are nonsingular, hence
This completes the proof.

The proof method of Lemma 4 is inspired by Marsaglia and Styan [5, Theorem 9]. By (4), we get the rank equality for the sum of finitely many idempotent matrices; it is different from the one of three idempotent matrices (3) given by Tian and Styan. Consequently, to find the generalization of Proposition 3 and solve the open problem given by Tian and Styan (see, e.g., [2]), it is necessary to seek a new method different from Lemma 4.

Lemma 5 (see [7, Problem 4.9]). *Let be idempotent. Then .*

In this section, from now on, for , one denotes

Theorem 6. *For any , let be idempotent. Then
*

*Proof. *From Lemma 4, it follows that

On the other hand, by block Gaussian elimination, we will see that

In fact, let us write the matrix
as the quadripartitioned matrix
where
By (11) and (13), it suffices to show
Direct calculations to (13) show that
where
If we define , by (14) and (17), we get
Moreover, let
then

By (14) and (19), we see that
Then by applying (14) and (19) yields
Thus,
Hence it follows from (14) and (19) that
Consequently, from (16)–(24), it follows thatwhere

Since , , and are nonsingular, we get
Also, it is easy to verify that

Since and are nonsingular, by (13) and (26)–(28), we obtain
Combining (13) with (29) together with Lemma 5 yields the desired results.

When , Theorem 6 leads to Proposition 3 at once, and when , it leads to Proposition 1; for the idempotent matrices and , it follows that

For the sum of two idempotent matrices, Tian and Styan have given out many rank equalities (see [1, Theorem 2.4] and [2, Theorems 2.1, 2.3, 2.4, and 2.7]). Let be idempotent; using Theorem 6 together with [6, Theorem 6] and [6, (26)] yields the equalities as follows:

Theorem 6 together with (31) and (32) indicates that the sum of idempotent matrices has various kinds of rank equalities, as shown in the discussions in the literature [1, 2].

In view of (11), by applying Lemma 5, we see the following.

Corollary 7. *For any , let be idempotent. Then
*

This immediately implies that the difference of the ranks of two block matrices in the left side of (33) is always equal to or , independently on the choice of , when .

#### 3. The Rank Formulas for the Sum of Scalar-Potent Matrices and Applications

Theorem 6 can easily be extended to scalar-potent matrices; in fact, So is idempotent.

Theorem 8. *For any given , let be scalar-potent (determined by ), . Then
*

*Proof. *By (8) and (34), we get

On the other hand, using (36), we obtain
with . Since is nonsingular, by (37), we can write
From (34), is idempotent. Using Theorem 6 together with (38) yields the equality
We note that
From (39) and (40), we get the desired result since .

When , this leads immediately to Proposition 2, since it can be written as with , , and .

For any given idempotent matrix , Farebrother and Trenkler [8] denoted the set of generalized quadratic matrices as If , it coincides with the definition of a quadratic matrix (see, e.g., [9]). In view of [10, Lemma 1] and [11, Lemma 2.2], we conclude that (42) can be expressed equivalently as If , then from (42) and (43), we see that

Lemma 9. *For any given idempotent matrix , if satisfies with , then is a scalar-potent matrix determined by .*

*Proof. *For the matrix , there exists a nonsingular matrix such that . From , we can write being . From , we get ; namely, . We have . It is seen from the fact that a matrix is diagonalizable if and only if its minimal polynomial has simple roots (see [12, Corollary 3.3.10]). Thus, there exists a nonsingular matrix such that . Let ; then is nonsingular and
Hence
Now, it is evident that .

Theorem 10. *For any given idempotent matrices and any , if satisfies with , , then
*

*Proof. *For the idempotent matrices , by applying Lemma 9, we see that is a scalar-potent matrix determined by ; then results follow from Theorem 8.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank Professor Y. Tian for his helpful comments and suggestions on this paper. The authors would also like to thank all referees for their patience in reading this paper and their valuable comments and suggestions that are helpful in improving and clarifying the paper. This work has been supported by the National Natural Science Foundation of China grant (61373140), the key item of Hercynian building for the colleges and universities service in Fujian Province (2008HX03), and the Special Scientific Research Program in Fujian Province Universities of China Grant (JK2013044).