#### Abstract

Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.

#### 1. Introduction

First, we use the following notations. Let stand for the set of complex matrices. The symbols , , , and represent the conjugate transpose, range, rank, and determinant of , respectively. The smallest positive integer such that is called the index of and it is denoted by . The Moore–Penrose inverse of is the unique matrix satisfying the following equations:and the unique matrix is denoted by [1, 2]. Furthermore, we denote

The Drazin inverse of is the unique matrix such that

and the unique matrix is usually denoted by , where [1, 2]. In particular, when , is called the group inverse of and is denoted by . Therefore, we call it a group invertible matrix with index 1. The symbol stands for the set of group invertible matrices in :

Baksalary and Trenkler [3] defined the core inverse of a complex matrix with index 1. Let ; the core inverse of is the unique matrix which satisfies the following equations:and it is denoted by . Subsequently, a variety of new generalized inverses have been established successively. Let with . The core-EP inverse of is the unique matrix satisfying , , and , and the core-EP inverse of is denoted by [4]; the B-T inverse of is the unique matrix satisfying , and the B-T inverse of is denoted by [5]; the DMP inverse of is the unique matrix satisfying , , and , and the DMP inverse of is denoted by [6]; the dual DMP inverse of is the unique matrix satisfying , , and , and the dual DMP inverse of is denoted by [6]; the CMP inverse of is the unique matrix satisfying , , and , and the CMP inverse of is denoted by [7]. It is easy to see that core-EP inverse and DMP inverse are both generalized core inverses, which are extensions of core inverse on square matrices without index constraint, and when , .

Furthermore, Wang and Chen [8] proposed a generalized group inverse. Let , if satisfies the following equations:where is called the WG inverse of , and is unique. It is usually denoted by . By applying the definition, we can obtain and . It is noteworthy that and , when . Then, Ferryra et al. [9] extended the definition of WG inverse to the general matrix, defined the weighted WG inverse, and gave its expression, properties, and characterizations; Mosić and Zhang [10] established the weighted WG inverse of Hilbert space operator; Zhou et al. [11] generalized WG inverse to a proper -ring and gave a new characterization of WG inverse; Zhou et al. [12] generalized m-WG inverse to a unitary ring with involution and gave some properties of m-WG inverse; Mosić and Stanimirović [13] gave new characterizations, limit representations, integral representations, and perturbation formulae of the WG inverse.

By applying the WG inverse, Wang and Liu [14] introduced the definition of WG matrix based on the properties and characterizations of WG inverse. Let ; if commutes with its WG inverse, is called WG matrix. The symbol stands for the set of WG matrices in [14]:

Subsequently, Yan et al. [15] investigated some new characterizations of weak group inverse by projection, the Bott–Duffin inverse, etc.

Matrix decomposition is very important, which not only functions as a significant role in every branch of mathematics but also has a wide range of applications in engineering. With the development of new generalized inverses, new research tools such as matrix decomposition and algorithm are also given. Wang established the core-EP decomposition of square matrix over complex fields [16]. Core-EP decomposition is one of the commonly used tools to study core-EP inverse and several new generalized inverses.

Lemma 1 (see [16], core-EP decomposition). *Let with . Then, there exist and , such that , where , , and . Furthermore,where is a unitary matrix, is nonsingular, and is nilpotent.**By applying the above decomposition, it is easy to verify*

Lemma 2 (see [14]). *Let with , then and*

Lemma 3 (see [8, 14]). *Let be as in the form (8). Among , , , and , any two of them are equivalent. If , thenwhere is a positive integer and is the index of .*

Lemma 4 (see [16–20]). *Let be as in the form [21], then*(1)*(2)**(3)**(4)**(5)**(6)**(7)**where , , , and .*

The classical Cayley–Hamilton theorem is one of the most important theorems in matrix theory. On the basis of the classical Cayley–Hamilton theorem, mathematicians established the rectangular matrix, block matrix, pair of block matrix, and other matrices as well as more generalized Cayley–Hamilton theorem for generalized inverse matrices. They also gave application of generalized Cayley–Hamilton theorem in several control systems [22–25]. In [26], Wang, Chen, and Yan gave the generalized Cayley–Hamilton theorem of core-EP inverse matrix and DMP inverse matrix by core-EP decomposition, and the characteristic polynomial equations of core-EP inverse matrix and DMP inverse matrix were also discussed. In [2, 27], the researchers studied the applications of generalized Cayley–Hamilton theorems in generalized inverses such as Drazin inverse and Moore–Penrose inverse. Based on the above researches, this paper will focus on the WG matrix, the equivalent characterizations of WG matrix, and the generalized Cayley–Hamilton theorem for special matrices including WG matrix.

#### 2. Some Characterizations of WG Matrix

In [14], the definition and characterizations of WG matrix are given through the commutativity of matrix and WG inverse. In [15], Yan et al. investigated some new characterizations of WG matrix.

Theorem 1 (see [15]). *Let with , then the following conditions are equivalent:*(1)*;*(2)*;*(3)*;*(4)*;*(5)*;*(6)*;*(7)*.*

It is pointed out that the set of group invertible matrices is a subset of WG matrices set. Special matrices such as WG matrix, group matrix, EP, -EP, and -EP matrix have rich intersection [14]. In this section, we will mainly apply core-EP decomposition to study the characterization of WG matrix.

Theorem 2. *Let with , then the following conditions are equivalent:*(1)*, where and are as in the form [21];*(2)*;*(3)*;*(4)*;*(5)* commutes with ;*(6)* is a WG matrix for any unitary matrix ;*(7)* commutes with ;*(8)* commutes with ;*(9)* = ;*(10)* commutes with ;*(11)* = ;*(12)* = ;*(13)*;*(14)*;*(15)*;*(16)*;*(17)*;*(18)*;*(19)*;*(20)*;*(21)*;*(22)* commutes with ;*(23)*;*(24)*;*(25)*;*(26)*;*(27)*;*(28)*;*(29)* commutes with ;*(30)*;*(31)*;*(32)*;*(33)*;*(34)*;*(35)*;*(36)*;*

*Proof. *From [14], we know that Conditions (1)–(5) are equivalent.

Let be a unitary matrix, then andThus, Conditions (2) and (6) are equivalent.

Let the core-EP decomposition of be as in the form [21]. By using Lemma 3, we obtainApplying [21, 27], we haveBy comparing [4, 7], we can getThus, Conditions (1) and (7) are equivalent.

By Lemma 4, we obtainBy applying [10, 13], we haveFrom what has been discussed above, we can surely come to the conclusion thatThus, Conditions (1) and (8) are equivalent.

Because of [7, 16], we can getThus, Conditions (1) and (9) are equivalent.

From [6, 23], we obtainBy comparing the above equations, we haveThus, Conditions (1) and (10) are equivalent.

By using [19], we can getBy applying the above equations, we obtainThus, Conditions (1) and (11) are equivalent.

Applying [10, 13], we haveBy comparing the above equations, we can getThus, Conditions (1) and (12) are equivalent.

By [19], we obtainFrom the above equation and [6], we can getThus, Conditions (1) and (13) are equivalent.

By applying [13, 23], we haveBy comparing the above equation and [6], we can getThus, Conditions (1) and (14) are equivalent.

Because of [6, 27], we obtainBy the above equation and [27], we haveThus, Conditions (1) and (15) are equivalent.

From [6, 27], we can getBy comparing the above equation and [6], we obtainHence, Conditions (1) and (16) are equivalent.

By using Lemma 2, we haveFrom the above equation and [23], we obtainThus, Conditions (1) and (17) are equivalent.

By applying Lemma 4, we can getBy (41), we obtainBy applying the above formula and (39), we have if and only if . Thus, Conditions (1) and (18) are equivalent.

By applying Lemma 4, we can getBecause of (43), we obtainBy comparing the above formula and (39), we obtain if and only if . Thus, Conditions (1) and (19) are equivalent.

Applying [6, 27], we haveIf , then . Conversely, let ; we get , that is, .

If the index of is equal to 1, then , that is, . If the index of is equal to 2, then . From , we have . Since is invertible, we obtain . Let the index of be more than or equal to 3, then and . If postmultiplication by , then . Since is invertible and , then . Furthermore, postmultiplication by , then . Since is invertible and , then . By repeating the process times, we have .

From what has been discussed above, if and only if . Thus, Conditions (1) and (20) are equivalent.

By [6, 21, 23], we obtainBy comparing the above equations, we can getThus, Conditions (1) and (21) are equivalent.

By applying (35) and (37), we have if and only if . Thus, Conditions (1) and (22) are equivalent.

Because of [19], we haveBy comparing the above equation and [7], we obtain . is equivalent to . Hence, Conditions (1) and (23) are equivalent.

From [2, 23], we have if and only if . Thus, Conditions (1) and (24) are equivalent.

By using Lemma 4, we obtainBy using (49), we haveBy comparing the above formula and [2], we can get if and only if . Thus, Conditions (1) and (25) are equivalent.

By Lemma 4, we obtainBy applying (51), we haveApplying the above formula and [2], we can get if and only if . Hence, Conditions (1) and (26) are equivalent.

Because of Lemma 4, we obtainFrom (53), we haveBy comparing the above formula and [2], we can get if and only if . Hence, Conditions (1) and (27) are equivalent.

By using Lemma 1 and Lemma 4, we obtainBy using the above formula, we haveIf , we can easily prove that . Conversely, if the index of is equal to 1 or 2, then obviously . Let the index of be more than or equal to 3; we get , that is, . Postmultiplying the above equation by , then , that is, . Through , we have . In the same way, we obtain , that is, . From what has been discussed above, we have if and only if . Hence, Conditions (1) and (28) are equivalent.

By Lemma 1 and Lemma 4, we can getBy applying [23] and (57), we haveSince the above formula, and are equivalent to . Thus, Conditions (1) and (29) are equivalent.

Because of [13] and (57), we can getIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (30) are equivalent.

From (59), we haveIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (31) are equivalent.

By using [13], (43), and (59), we can getIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Thus, Conditions (1) and (32) are equivalent.

Applying Lemma 4, [10], and (59), we obtainIf , then . Conversely, let , then and . By using the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (33) are equivalent.

By Lemma 4, [13], and (59), we haveIf , then . Conversely, let , then and . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (34) are equivalent.

By applying [4, 8], we obtain if and only if . Hence, Conditions (1) and (35) are equivalent.

Because of [16], (57) andIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (36) are equivalent.

#### 3. Generalized Cayley–Hamilton Theorem

In this section, we extend the classical Cayley–Hamilton theorem to some special matrix such as the WG matrix.

Theorem 3 (see [21]). *Let ; the characteristic polynomial of isthen**In [21], if is singular, then .*

Theorem 4. *Let be singular with . Ifthenwhere is the weak group inverse of the matrix .*

*Proof. *Let be singular; we use the Cayley–Hamilton theorem, thenPostmultiplying the above equation by , we getBy using the properties of the WG matrix, that is, , we have . A similar method can be used to obtain , , , and . Substituting the above equations into (71), we have (69).

*Example 1. *LetIt is easy to confirm that the weak group inverse isThen,From the classical Cayley–Hamilton theorem, we haveBy applying Theorem 4, we obtainNext, we extend the classical Cayley–Hamilton theorem to the WG inverse matrix. Let with . From Lemma 1 and Lemma 2, it can be obtainedThe characteristic polynomial of isIt is given by the classical Cayley–Hamilton theoremPostmultiplying the above equation by ,By using (77) and (78), we have Theorem 5.

Theorem 5. *Let with , then the characteristic polynomial of iswhere is as in (78).*

*Example 2. *Letthen