The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem

First, we use the following notations. Let Cm,n stand for the set of m × n complex matrices. .e symbols A∗, R(A), rk(A), and det(A) represent the conjugate transpose, range, rank, and determinant of A, respectively. .e smallest positive integer k such that rk(Ak+1) � rk(A) is called the index of A ∈ Cm,n and it is denoted by Ind(A). .e Moore–Penrose inverse of A ∈ Cm,n is the unique matrix X ∈ Cn,m satisfying the following equations:


Introduction
First, we use the following notations. Let C m,n stand for the set of m × n complex matrices. e symbols A * , R(A), rk(A), and det(A) represent the conjugate transpose, range, rank, and determinant of A, respectively. e smallest positive integer k such that rk(A k+1 ) � rk(A k ) is called the index of A ∈ C m,n and it is denoted by Ind(A). e Moore-Penrose inverse of A ∈ C m,n is the unique matrix X ∈ C n,m satisfying the following equations: (1) and the unique matrix X is denoted by X � A † [1,2]. Furthermore, we denote e Drazin inverse of A ∈ C n,n is the unique matrix X ∈ C n,n such that and the unique matrix X is usually denoted by X � A D , where k � Ind(A) [1,2]. In particular, when k � 1, X is called the group inverse of A and is denoted by X � A # . erefore, we call it a group invertible matrix with index 1. e symbol C CM n stands for the set of group invertible matrices in C n,n : Baksalary and Trenkler [3] defined the core inverse of a complex matrix with index 1. Let A ∈ C CM n ; the core inverse of A is the unique matrix which satisfies the following equations: and (AX) * � AX, and the core-EP inverse of A is denoted by A ○ † [4]; the B-T inverse of A is the unique matrix X ∈ C n,n satisfying X � (AE A ) † , and the B-T inverse of A is denoted by A ◇ [5]; the DMP inverse of A is the unique matrix X ∈ C n,n satisfying XAX � X, XA � A D A, and A k X � A k A ○ † , and the DMP inverse of A is denoted by X � A D, † [6]; the dual DMP inverse of A is the unique matrix X ∈ C n,n satisfying XAX � X, AX � AA D , and XA k � A ○ † A k , and the dual DMP inverse of A is denoted by A †,D [6]; the CMP inverse of A ∈ C n,n is the unique matrix X ∈ C n,n satisfying XAX � X, AX � AA D AA ○ † , and XA � A ○ † AA D A, and the CMP inverse of A ∈ C n,n is denoted by A C, † [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 A ∈ C CM n , if X satisfies the following equations: where X is called the WG inverse of A, and X is unique. It is usually denoted by X � A Ⓦ . By applying the definition, we can obtain n . en, 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 A ∈ C n,n ; if A commutes with its WG inverse, A is called WG matrix. e symbol C WG n stands for the set of WG matrices in C n,n [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 A ∈ C n,n with Ind(A) � k. en, there exist A 1 and A 2 , such that A � where U is a unitary matrix, T ∈ C rk(A k ),rk(A k ) is nonsingular, and N is nilpotent. By applying the above decomposition, it is easy to verify Lemma 2 (see [14]).
Among A ∈ C WG n , (A 2 ) Ⓦ � (A Ⓦ ) 2 , SN � 0, and AA Ⓦ � A Ⓦ A, any two of them are equivalent. If A ∈ C WG n , then where t is a positive integer and k is the index of A.
Lemma 4 (see [16][17][18][19][20]). Let A ∈ C n,n be as in the form [21], then e 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. ey also gave application of generalized Cayley-Hamilton theorem in several control systems [22][23][24][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.

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 A ∈ C n,n with Ind(A) � k, then the following conditions are equivalent: 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, i-EP, and k-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 A ∈ C n,n with Ind(A) � k, then the following conditions are equivalent: (1) SN � 0, where S and N are as in the form [21]; Proof. From [14], we know that Conditions (1)-(5) are equivalent.
Let U be a unitary matrix, then (UAU * ) Ⓦ � UA Ⓦ U * and us, Conditions (2) and (6) are equivalent. Let the core-EP decomposition of A be as in the form [21]. By using Lemma 3, we obtain Journal of Mathematics Applying [21,27], we have By comparing [4,7], we can get us, Conditions (1) and (7) are equivalent. By Lemma 4, we obtain By applying [10,13], we have From what has been discussed above, we can surely come to the conclusion that us, Conditions (1) and (8) are equivalent. Because of [7,16], we can get us, Conditions (1) and (9) are equivalent. From [6,23], we obtain By comparing the above equations, we have us, Conditions (1) and (10) are equivalent. By using [19], we can get By applying the above equations, we obtain us, Conditions (1) and (11) are equivalent. Applying [10,13], we have 4 Journal of Mathematics By comparing the above equations, we can get us, Conditions (1) and (12) are equivalent. By [19], we obtain From the above equation and [6], we can get us, Conditions (1) and (13) are equivalent. By applying [13,23], we have By comparing the above equation and [6], we can get us, Conditions (1) and (14) are equivalent. Because of [6,27], we obtain By the above equation and [27], we have us, Conditions (1) and (15) are equivalent. From [6,27], we can get By comparing the above equation and [6], we obtain Hence, Conditions (1) and (16) are equivalent. By using Lemma 2, we have From the above equation and [23], we obtain us, Conditions (1) and (17) are equivalent. By applying Lemma 4, we can get By (41), we obtain

Journal of Mathematics
Because of (43), we obtain By comparing the above formula and (39), we obtain SN � 0 if and only if A(A 2 ) Ⓦ � AA D, † A Ⓦ . us, Conditions (1) and (19) are equivalent.
Applying [6,27], we have If the index of A is equal to 1, then N � 0, that is, SN � 0. If the index of A is equal to 2, then N 2 � 0. From TSN + SN 2 � 0, we have TSN � 0. Since T is invertible, we obtain SN � 0. Let the index of A be more than or equal to 3, then N k � 0 and N k− 1 ≠ 0. If TSN + SN 2 � 0 postmultiplication by N k− 2 , then TSN k− 1 + SN k � 0. Since T is invertible and N k � 0, then SN k− 1 � 0. Furthermore, TSN + SN 2 � 0 postmultiplication by N k− 3 , then TSN k− 2 + SN k− 1 � 0. Since T is invertible and N k− 1 � 0, then SN k− 2 � 0. By repeating the process k − 1 times, we have SN � 0.
Because of Lemma 4, we obtain From (53), we have By comparing the above formula and [2], we can get SN � 0 if and only if (A Ⓦ ) 2 A � AA C, † A Ⓦ . Hence, Conditions (1) and (27) are equivalent.
By using Lemma 1 and Lemma 4, we obtain By using the above formula, we have If SN � 0, we can easily prove that A Ⓦ A D A � A Ⓦ . Conversely, if the index of A is equal to 1 or 2, then obviously SN � 0. Let the index of A be more than or equal to 3; we get Postmultiplying the above equation by N k− 2 , then TSN k+k− 3 + · · · + T k− 1 SN k− 1 � 0, that is, SN k− 1 � 0. rough TSN k− 1 + · · · + T k− 1 SN � 0, we have T 2 SN k− 2 + · · · + T k− 1 SN � 0. In the same way, we obtain T k− 1 SN � 0, that is, SN � 0. From what has been discussed above, we have SN � 0 if and only if A Ⓦ A D A � A Ⓦ . Hence, Conditions (1) and (28) are equivalent.
By Lemma 1 and Lemma 4, we can get By applying [23] and (57), we have Since the above formula, us, Conditions (1) and (29) are equivalent.
Because of [13] and (57), we can get If By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain SN � 0. From what has been discussed above, SN � 0 is equivalent to AA D (I − AA ○ † )A � 0. Hence, Conditions (1) and (30) are equivalent.
From (59), we have Journal of Mathematics , that is, TSN k− 1 + · · · + T k− 1 SN � 0. By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain SN � 0. From what has been discussed above, SN � 0 is equivalent to A k (I − AA ○ † )A � 0. Hence, Conditions (1) and (31) are equivalent.
By using [13], (43), and (59), we can get By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain SN � 0. From what has been discussed above, SN � 0 is equivalent to AA D, † (I − AA ○ † )A � 0. us, Conditions (1) and (32) are equivalent.

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

Theorem 4.
Let A ∈ C n,n be singular with Ind(A) � k. If det sI n − A � s n + a n− 1 s n− 1 + · · · + a 1 s, then A Ⓦ + a n− 1 A Ⓦ 2 + · · · + a 1 A Ⓦ n � 0, where A Ⓦ ∈ C n,n is the weak group inverse of the matrix A.
Proof. Let A ∈ C n,n be singular; we use the Cayley-Hamilton theorem, then A n + a n− 1 A n− 1 + · · · + a 1 A � 0.
Postmultiplying the above equation by (A Ⓦ ) n+1 , we get A n A Ⓦ n+1 + a n− 1 A n− 1 A Ⓦ n+1 generalized Cayley-Hamilton theorem for special matrices such as WG matrix is given. We are convinced that researches about the WG matrix will also gain more attention in the near future. Some perspectives for further researches can be described as follows: (1) our further goal will be to investigate the definition, properties, and characterizations of tensor WG matrix; (2) further, we will study the iterative algorithm of the tensor WG matrix and its application [28].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.