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 P 1 + ⋯+P k with P 1,…, P k be idempotent (k > 3) 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.


Introduction
Let C × and (C) 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 Z + be the set of all the positive integer numbers. The symbols ( ) and stand for the rank and transpose of ∈ C × , respectively, while tr denotes the trace of a square matrix . A matrix ∈ C × is said to be , if 2 = , and -(determined by ), if 2 = , for some (0 ̸ = ) ∈ C (see, e.g., [1]). When = 1, 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][2][3][4][5][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.

2
The Scientific World Journal show that establishing various kinds of rank equalities for idempotent matrices is interesting. Tian and Styan pointed out that rank equalities for the sum 1 +⋅ ⋅ ⋅+ with 1 , . . . , be idempotent ( > 3) 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.

Main Results
Before showing main results, we need some preparations.
for any ∈ Z + .
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.
In this section, from now on, for 1 , 2 , . . . , ∈ C × , one denotes The Scientific World Journal 3

Proof. From Lemma 4, it follows that
) . (10) On the other hand, by block Gaussian elimination, we will see that In fact, let us write the matrix as the quadripartitioned matrix By (11) and (13), it suffices to show If we define 11 = 11 + 12 21 , by (14) and (17), we get By (14) and (19), we see that Then by applying (14) and (19) yields Hence it follows from (14) and (19) that Since  ) .
Also, it is easy to verify that Combining (13) with (29) together with Lemma 5 yields the desired results.
When = 3, Theorem 6 leads to Proposition 3 at once, and when = 2, it leads to Proposition 1; for the idempotent matrices and , it follows that   [6, (26)] yields the equalities as follows: Theorem 6 together with (31) and (32) indicates that the sum of (≥ 3) 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 ∈ Z + , let 1 , . . . , ∈ C × 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 ( 1 ) or tr 1 , 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 (1/ ) is idempotent.
When = 2, this leads immediately to Proposition 2, since it can be written as with 2 = , 2 = , and ̸ = 0. 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] If ∈ Ω ( ), then from (42) and (43), we see that The Scientific World Journal 7 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.