Research Article | Open Access
On the Open Problem Related to Rank Equalities for the Sum of Finitely Many Idempotent Matrices and Its Applications
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.
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., ). 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.
Tian and Styan have extended the rank equality for the sum of idempotent matrices to the scalar-potent matrices (see, e.g., ).
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  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., ). 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 .
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., ), 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.
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:
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  denoted the set of generalized quadratic matrices as If , it coincides with the definition of a quadratic matrix (see, e.g., ). 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
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).
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