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

Authors: Na Liu; Hongxing Wang
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 1. Introduction

First, we use the following notations. Letℂm,n stand for the set of m×n complex matrices. The symbols A∗, ℛA, rkA, and detA represent the conjugate transpose, range, rank, and determinant of A, respectively. The smallest positive integer k such that rkAk+1=rkAk is called the index of A∈ℂm,n and it is denoted by IndA. The Moore–Penrose inverse of A∈ℂm,n is the unique matrix X∈ℂn,m satisfying the following equations:(1)AXA=A,XAX=X,AX∗=AX,XA∗=XA,and the unique matrix X is denoted by X=A† [1, 2]. Furthermore, we denote(2)EA=AA†,FA=In−A†A.The Drazin inverse ofA∈ℂn,n is the unique matrix X∈ℂn,n such that(3)AkXA=Ak,XAX=X,AX=XA,and the unique matrixX is usually denoted by X=AD, where k=IndA [1, 2]. In particular, when k=1, X is called the group inverse of A and is denoted by X=A#. Therefore, we call it a group invertible matrix with index 1. The symbol ℂnCM stands for the set of group invertible matrices in ℂn,n:(4)ℂnCM=ArkA2=rkA,A∈ℂn,n.Baksalary and Trenkler [3] defined the core inverse of a complex matrix with index 1. Let A∈ℂnCM; the core inverse of A is the unique matrix which satisfies the following equations:(5)AX=AA○†,ℛX⊆ℛA,and it is denoted by X=A○#. Subsequently, a variety of new generalized inverses have been established successively. Let A∈ℂn,n with IndA=k. The core-EP inverse of A is the unique matrix X∈ℂn,n satisfying XAk+1=Ak, XAX=X, 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∈ℂn,n satisfying X=AEA†, and the B-T inverse of A is denoted by A⋄ [5]; the DMP inverse of A is the unique matrix X∈ℂn,n satisfying XAX=X, XA=ADA, and AkX=AkA○†, and the DMP inverse of A is denoted by X=AD,† [6]; the dual DMP inverse of A is the unique matrix X∈ℂn,n satisfying XAX=X, AX=AAD, and XAk=A○†Ak, and the dual DMP inverse of A is denoted by A†,D [6]; the CMP inverse of A∈ℂn,n is the unique matrix X∈ℂn,n satisfying XAX=X, AX=AADAA○†, and XA=A○†AADA, and the CMP inverse of A∈ℂn,n is denoted by AC,† [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 A∈ℂnCM, A○#=AD,†=A○†.Furthermore, Wang and Chen [8] proposed a generalized group inverse. Let A∈ℂnCM, if X satisfies the following equations:(6)AX2=X,AX=A○†A,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 AⓌAAⓌ=AⓌ and ℛAⓌ⊆ℛAk. It is noteworthy that AⓌ=A# and A#≠A#, when A∈ℂnCM. 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 A∈ℂn,n; if A commutes with its WG inverse, A is called WG matrix. The symbol ℂnWG stands for the set of WG matrices in ℂn,n [14]:(7)ℂnWG=AAAⓌ=AⓌA,A∈ℂn,n.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).
LetA∈ℂn,n with IndA=k. Then, there exist A1 and A2, such that A=A1+A2, where A1∈ℂnCM, A2k=0, and A1∗A2=A2A1=0. Furthermore,(8)A1=UTS00U∗,A2=U000NU∗,where U is a unitary matrix, T∈ℂrkAk,rkAk is nonsingular, and N is nilpotent.
By applying the above decomposition, it is easy to verify(9)AkAk○†=UIrkAk000U∗.Lemma 2 (see [14]).
LetA∈ℂn,n with IndA=k, then rkAⓌ=rkAD=rkAk and(10)AⓌ=A1#=UT−1T−2S00U∗,=AA○†A#=A○†2A=A2○†A=AkAk+2○#A=A2EAk○†A.Lemma 3 (see [8, 14]).
LetA∈ℂn,n be as in the form (8). Among A∈ℂnWG, A2Ⓦ=AⓌ2, SN=0, and AAⓌ=AⓌA, any two of them are equivalent. If A∈ℂnWG, then(11)AⓌ=AD=Ak+1ⓌAk=At+1○†At,where t is a positive integer and k is the index of A.Lemma 4 (see [16–20]).
LetA∈ℂn,n be as in the form [21], then(1)
A○†=UT∗Δ−T∗ΔSN○†FNS∗ΔN○†−FNS∗ΔSN○†U∗;(2)
AD=UT−1T−k+1T˜00U∗;(3)
A○†=UT−1000U∗;(4)
A⋄=UT∗△1−T∗△1SN⋄EN−EN⋄S∗△1N⋄−EN−EN⋄S∗△1SN⋄U∗;(5)
AD,†=UT−1T−k+1T˜NN○†00U∗;(6)
A†,D=UT∗△T∗△T−kT˜FNS∗△FNS∗△T−kT˜U∗;(7)
AC,†=UT∗△T∗△T−kT˜NN○†FNS∗△FNS∗△T−kT˜NN○†U∗,where △1=TT∗+SEN−EN⋄S∗−1, k=IndA, T˜=∑i=0k−1TiSNk−1−i, and △=TT∗+SFNS∗−1.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]).
LetA∈ℂn,n with IndA=k, then the following conditions are equivalent:(1)
A∈ℂnWG;(2)
AkA○†=AkA○†;(3)
AkA○†=AkAD,†;(4)
A○†Ak=AkAⓌ;(5)
AAD=A○†A;(6)
AAD=AAⓌ;(7)
A○†A=AD,†A.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.
LetA∈ℂn,n with IndA=k, then the following conditions are equivalent:(1)
SN=0, where S and N are as in the form [21];(2)
AAⓌ=AⓌA;(3)
A2Ⓦ=AⓌ2;(4)
AⓌ=AD;(5)
A○†A commutes with A○†A2;(6)
UAU∗ is a WG matrix for any unitary matrix U;(7)
A commutes with AAⓌ;(8)
A commutes with A○†A;(9)
AA○†A = AAⓌA;(10)
AⓌ commutes with AⓌA;(11)
AkAk○†A2 = AkAk○†A2;(12)
AA○†A2 = AA○†A2;(13)
AⓌA=AⓌAkAk○†A;(14)
AⓌA=AⓌAA○†A;(15)
AAⓌ2A=AAⓌ;(16)
AⓌA2AⓌ=AⓌA;(17)
AA2Ⓦ=AⓌ;(18)
AA2Ⓦ=AA○†AⓌ;(19)
AA2Ⓦ=AAD,†AⓌ;(20)
AAⓌ2=AⓌ2AAⓌA2;(21)
AⓌA2=A2AⓌ2;(22)
AAⓌ commutes with AⓌA;(23)
AAⓌA=AkAk○†A;(24)
AⓌ2A=AⓌ;(25)
AⓌ2A=AD,†AAⓌ;(26)
AⓌ2A=AA†,DAⓌ;(27)
AⓌ2A=AAC,†AⓌ;(28)
AⓌADA=AⓌ;(29)
AⓌ commutes with AAD;(30)
AADI−AA○†A=0;(31)
AkI−AA○†A=0;(32)
AAD,†I−AA○†A=0;(33)
A†,DI−AA○†A=0;(34)
AC,†I−AA○†A=0;(35)
A○†A2=A2AⓌ;(36)
A2AD=AA○†A;Proof.
From [14], we know that Conditions (1)–(5) are equivalent.
LetU be a unitary matrix, then UAU∗Ⓦ=UAⓌU∗ and(12)UAAⓌU∗=UAU∗UAⓌU∗=UAU∗UAU∗Ⓦ,UAⓌAU∗=UAⓌU∗UAU∗=UAU∗ⓌUAU∗.
Thus, Conditions (2) and (6) are equivalent.
Let the core-EP decomposition ofA be as in the form [21]. By using Lemma 3, we obtain(13)AAⓌ=UTS0NT−1T−2S00U∗=UIrkAkT−1S00U∗,(14)AⓌA=UT−1T−2S00TS0NU∗=UIrkAkT−1S+T−2SN00U∗.
Applying [21, 27], we have(15)AAAⓌ=UTS0NIrkAkT−1S00U∗=UTS00U∗,(16)AAⓌA=UIrkAkT−1S00TS0NU∗=UTS+T−1SNS00U∗.
By comparing [4, 7], we can get(17)SN=0⟺AAAⓌ=AAⓌA.
Thus, Conditions (1) and (7) are equivalent.
By Lemma4, we obtain(18)AA○†=UTS0NT−1000U∗=UIrkAk000U∗,(19)A○†A=UT−1000TS0NU∗=UIrkAkT−1S00U∗.
By applying [10, 13], we have(20)AA○†A=UIrkAk000TS0NU∗=UTS00U∗,(21)A○†AA=UIrkAkT−1S00TS0NU∗=UTS+T−1SN00U∗.
From what has been discussed above, we can surely come to the conclusion that(22)SN=0⟺AA○†A=A○†AA.
Thus, Conditions (1) and (8) are equivalent.
Because of [7, 16], we can get(23)SN=0⟺AA○†A=AAⓌA.
Thus, Conditions (1) and (9) are equivalent.
From [6, 23], we obtain(24)AⓌAⓌA=UT−1T−2S00IrkAkT−1S+T−2SN00U∗=UT−1T−2S+T−3SN00U∗,(25)AⓌAAⓌ=UIrkAkT−1S+T−2SN00T−1T−2S00U∗=UT−1T−2S00U∗.
By comparing the above equations, we have(26)SN=0⟺AⓌAⓌA=AⓌAAⓌ.
Thus, Conditions (1) and (10) are equivalent.
By using [19], we can get(27)AkAk○†A2=UIrkAk000TS0NTS0NU∗=UT2TS+SN00U∗,AkAk○†A2=UIrkAk000TS0NIrkAk000TS0NU∗=UT2TS00U∗.
By applying the above equations, we obtain(28)SN=0⟺AkAk○†A2=AkAk○†A2.
Thus, Conditions (1) and (11) are equivalent.
Applying [10, 13], we have(29)AA○†A2=UIrkAk000TS0NTS0NU∗=UT2TS+SN00U∗,AA○†A2=UTS0NIrkAkT−1S00TS0NIrkAkT−1S00U∗=UT2TS00U∗.
By comparing the above equations, we can get(30)SN=0⟺AA○†A2=AA○†A2.
Thus, Conditions (1) and (12) are equivalent.
By [19], we obtain(31)AⓌAkAk○†A=UT−1T−2S00IrkAk000TS0NU∗=UIrkAkT−1S00U∗.
From the above equation and [6], we can get(32)SN=0⟺AWA=AWAkAk○†A.
Thus, Conditions (1) and (13) are equivalent.
By applying [13, 23], we have(33)AⓌAA○†A=UT−1T−2S00IrkAk000TS0NU∗=UIrkAkT−1S00U∗.
By comparing the above equation and [6], we can get(34)SN=0⟺AⓌA=AⓌAA○†A.
Thus, Conditions (1) and (14) are equivalent.
Because of [6, 27], we obtain(35)AAⓌ2A=UIrkAkT−1S00IrkAkT−1S+T−2SN00U∗=UIrkAkT−1S+T−2SN00U∗.
By the above equation and [27], we have(36)SN=0⟺AAⓌ2A=AAⓌ.
Thus, Conditions (1) and (15) are equivalent.
From [6, 27], we can get(37)AⓌA2AⓌ=UIrkAkT−1S+T−2SN00IrkAkT−1S00U∗=UIrkAkT−1S00U∗.
By comparing the above equation and [6], we obtain(38)SN=0⟺AⓌA2AⓌ=AⓌA.
Hence, Conditions (1) and (16) are equivalent.
By using Lemma2, we have(39)AA2W=UTS0NT−2T−4TS+SN00U∗=UT−1T−2S+T−3SN00U∗.
From the above equation and [23], we obtain(40)SN=0⟺AA2W=AW.
Thus, Conditions (1) and (17) are equivalent.
By applying Lemma4, we can get(41)AA○†=UTS0NT∗△−T∗△SN○†FNS∗△N○†−FNS∗△SN○†U∗=UIrkAk00NN○†U∗.
By (41), we obtain(42)AA○†AW=UIrkAk00NN○†T−1T−2S00U∗=UT−1T−2S00U∗.
By applying the above formula and (39), we have SN=0 if and only if AA2Ⓦ=AA○†AⓌ. Thus, Conditions (1) and (18) are equivalent.
By applying Lemma4, we can get(43)AAD,†=UTS0NT−1T−k+1T˜NN○†00U∗=UIrkAkT−kT˜NN○†00U∗.
Because of (43), we obtain(44)AAD,†AⓌ=UIrkAkT−kT˜NN○†00T−1T−2S00U∗=UT−1T−2S00U∗.
By comparing the above formula and (39), we obtain SN=0 if and only if AA2Ⓦ=AAD,†AⓌ. Thus, Conditions (1) and (19) are equivalent.
Applying [6, 27], we have(45)AAⓌ2=UIrkAkT−1S002U∗=UIrkAkT−1S00U∗,AⓌ2AAⓌA2=UT−1T−2S002IrkAkT−1S00TS0N2U∗,=UIrkAkT−1S+T−2SN+T−3SN200U∗.
IfSN=0, then AAⓌ2=AⓌ2AAⓌA2. Conversely, let AAⓌ2=AⓌ2AAⓌA2; we get T−1S+T−2SN+T−3SN2=T−1S, that is, TSN+SN2=0.
If the index ofA is equal to 1, then N=0, that is, SN=0. If the index of A is equal to 2, then N2=0. From TSN+SN2=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 Nk=0 and Nk−1≠0. If TSN+SN2=0 postmultiplication by Nk−2, then TSNk−1+SNk=0. Since T is invertible and Nk=0, then SNk−1=0. Furthermore, TSN+SN2=0 postmultiplication by Nk−3, then TSNk−2+SNk−1=0. Since T is invertible and Nk−1=0, then SNk−2=0. By repeating the process k−1 times, we have SN=0.
From what has been discussed above,SN=0 if and only if AAⓌ2=AⓌ2AAⓌA2. Thus, Conditions (1) and (20) are equivalent.
By [6, 21, 23], we obtain(46)AⓌA2=UIrkAkT−1S+T−2SN002U∗=UIrkAkT−1S+T−2SN00U∗,A2AⓌ2=UTS0N2T−1T−2S002U∗=UIrkAkT−1S00U∗.
By comparing the above equations, we can get(47)SN=0⟺AⓌA2=A2AⓌ2.
Thus, Conditions (1) and (21) are equivalent.
By applying (35) and (37), we have SN=0 if and only if AAWAWA=AWAAAW. Thus, Conditions (1) and (22) are equivalent.
Because of [19], we have(48)AkAk○†A=UIrkAk000TS0NU∗=UTS00U∗.
By comparing the above equation and [7], we obtain AAⓌA=AkAk○†A. SN=0 is equivalent to AAⓌA=AkAk○†A. Hence, Conditions (1) and (23) are equivalent.
From [2, 23], we have SN=0 if and only if AⓌ2A=AⓌ. Thus, Conditions (1) and (24) are equivalent.
By using Lemma4, we obtain(49)AD,†A=UT−1T−k+1T˜NN○†00TS0NU∗=UIrkAkT−1S+T−k+1T˜N00U∗.
By using (49), we have(50)AD,†AAⓌ=UIrkAkT−1S+T−k+1T˜N00T−1T−2S00U∗=UT−1T−2S00U∗=AⓌ.
By comparing the above formula and [2], we can get SN=0 if and only if AⓌ2A=AD,†AAⓌ. Thus, Conditions (1) and (25) are equivalent.
By Lemma4, we obtain(51)AA†,D=UTS0NT∗△T∗△T−kT˜FNS∗△FNS∗△T−kT˜U∗=UIrkAkT−kT˜00U∗.
By applying (51), we have(52)AA†,DAⓌ=UIrkAkT−kT˜00T−1T−2S00U∗=UT−1T−2S00U∗=AⓌ.
Applying the above formula and [2], we can get SN=0 if and only if AⓌ2A=AA†,DAⓌ. Hence, Conditions (1) and (26) are equivalent.
Because of Lemma4, we obtain(53)AAC,†=UTS0NT∗△T∗△T−kT˜NN○†FNS∗△FNS∗△T−kT˜NN○†U∗=UIrkAkT−kT˜NN○†00U∗.
From (53), we have(54)AAC,†AⓌ=UIrkAkT−kT˜NN○†00T−1T−2S00U∗=UT−1T−2S00U∗=AⓌ.
By comparing the above formula and [2], we can get SN=0 if and only if AⓌ2A=AAC,†AⓌ. Hence, Conditions (1) and (27) are equivalent.
By using Lemma1 and Lemma 4, we obtain(55)ADA=UT−1T−k+1T˜00TS0NU∗=UIrkAkT−1S+T−k+1T˜N00U∗.
By using the above formula, we have(56)AⓌADA=UT−1T−2S00IrkAkT−1S+T−k+1T˜N00U∗=UT−1T−2S+T−k+2T˜N00U∗.
IfSN=0, we can easily prove that AⓌADA=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 T−2S+T−k+2∑i=0k−1TiSNk−1−iN=T−2S, that is, TSNk−1+⋯+Tk−1SN=0. Postmultiplying the above equation by Nk−2, then TSNk+k−3+⋯+Tk−1SNk−1=0, that is, SNk−1=0. Through TSNk−1+⋯+Tk−1SN=0, we have T2SNk−2+⋯+Tk−1SN=0. In the same way, we obtain Tk−1SN=0, that is, SN=0. From what has been discussed above, we have SN=0 if and only if AⓌADA=AⓌ. Hence, Conditions (1) and (28) are equivalent.
By Lemma1 and Lemma 4, we can get(57)AAD=UTS0NT−1T−k+1T˜00U∗=UIrkAkT−kT˜00U∗.
By applying [23] and (57), we have(58)AADAⓌ=UIrkAkT−kT˜00T−1T−2S00U∗=UT−1T−2S00U∗=AⓌ.
Since the above formula,AⓌAAD=AⓌADA and AⓌAAD=AⓌ are equivalent to AⓌAAD=AADAⓌ. Thus, Conditions (1) and (29) are equivalent.
Because of [13] and (57), we can get(59)I−AA○†=UIrkAk00In−rkAkIrkAk000U∗=U000In−rkAkU∗,(60)AADI−AA○†A=UTkT−kT˜00000In−rkAkTS0NU∗=U0T−kT˜N00U∗.
IfSN=0, then AADI−AA○†A=0. Conversely, let AADI−AA○†A=0, then T−k∑i=0k−1TiSNk−1−iN=0, that is, TSNk−1+⋯+Tk−1SN=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 AADI−AA○†A=0. Hence, Conditions (1) and (30) are equivalent.
From (59), we have(61)AkI−AA○†A=UTkT˜00000In−rkAkTS0NU∗=U0T˜N00U∗.
IfSN=0, then AkI−AA○†A=0. Conversely, let AkI−AA○†A=0, then ∑i=0k−1TiSNk−1−iN=0, that is, TSNk−1+⋯+Tk−1SN=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 AkI−AA○†A=0. Hence, Conditions (1) and (31) are equivalent.
By using [13], (43), and (59), we can get(62)AAD,†I−AA○†A=UIrkAkT−kT˜NN○†00000In−rkAkTS0NU∗=U0T−kT˜N00U∗.
IfSN=0, then AAD,†I−AA○†A=0. Conversely, let AAD,†I−AA○†A=0, then T−k∑i=0k−1TiSNk−1−iN=0, that is, TSNk−1+⋯+Tk−1SN=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 AAD,†I−AA○†A=0. Thus, Conditions (1) and (32) are equivalent.
Applying Lemma4, [10], and (59), we obtain(63)A†,DI−AA○†A=UT∗△T∗△T−kT˜FNS∗△FNS∗△T−kT˜000In−rkAkTS0NU∗=U0T∗△T−kT˜N0FNS∗△T−kT˜NU∗.
IfSN=0, then A†,DI−AA○†A=0. Conversely, let A†,DI−AA○†A=0, then T∗TT∗+SFNS∗−1T−k∑i=0k−1TiSNk−1−iN=0 and FNS∗TT∗+SFNS∗−1T−k∑i=0k−1TiSNk−1−iN=0. By using 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†,DI−AA○†A=0. Hence, Conditions (1) and (33) are equivalent.
By Lemma4, [13], and (59), we have(64)AC,†I−AA○†A=UT∗△T∗△T−kT˜NN○†FNS∗△FNS∗△T−kT˜NN○†000In−rkAkTS0NU∗=U0T∗△T−kT˜N0FNS∗△T−kT˜NU∗.
IfSN=0, then AC,†I−AA○†A=0. Conversely, let AC,†I−AA○†A=0, then T∗TT∗+SFNS∗−1T−k∑i=0k−1TiSNk−1−iN=0 and FNS∗TT∗+SFNS∗−1T−k∑i=0k−1TiSNk−1−iN=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 AC,†I−AA○†A=0. Hence, Conditions (1) and (34) are equivalent.
By applying [4, 8], we obtain SN=0 if and only if A○†A2=A2AW. Hence, Conditions (1) and (35) are equivalent.
Because of [16], (57) and(65)AAAD=UTS0NIrkAkT−kT˜00U∗=UTT−k−1T˜00U∗.
IfSN=0, then A2AD=AA○†A. Conversely, let A2AD=AA○†A, then T−k−1∑i=0k−1TiSNk−1−i=S, that is, T−k−1SNk−1+⋯+T−1SN=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 A2AD=AA○†A. 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]).
LetA∈ℂn,n; the characteristic polynomial of A is(66)pAs=detsIn−A=sn+an−1sn−1+⋯+a1s+a0,then(67)pAA=An+an−1An−1+⋯+a1A+a0In=0.
In [21], if A is singular, then a0=0.Theorem 4.
LetA∈ℂn,n be singular with IndA=k. If(68)detsIn−A=sn+an−1sn−1+⋯+a1s,then(69)AⓌ+an−1AⓌ2+⋯+a1AⓌn=0,where AⓌ∈ℂn,n is the weak group inverse of the matrix A.Proof.
LetA∈ℂn,n be singular; we use the Cayley–Hamilton theorem, then(70)An+an−1An−1+⋯+a1A=0.
Postmultiplying the above equation byAⓌn+1, we get(71)AnAⓌn+1+an−1An−1AⓌn+1+⋯+a1AAⓌn+1=0.
By using the properties of the WG matrix, that is,AAⓌ2=AⓌ, we have AAⓌn+1=AAⓌ2AⓌn−1=AⓌAⓌn−1=AⓌn. A similar method can be used to obtain A2AⓌn+1=AⓌn−1, ⋯, An−1AⓌn+1=AⓌ2, and AnAWn+1=AW. Substituting the above equations into (71), we have (69).Example 1.
Let(72)A=10−10.
It is easy to confirm that the weak group inverseAW∈ℂn,n is(73)AⓌ=10−10.
Then,(74)detsI2−A=s−101s=s2−s.
From the classical Cayley–Hamilton theorem, we have(75)A2−A=10−102−10−10=0000.
By applying Theorem4, we obtain(76)AⓌ−AⓌ2=10−10−10−102=0000.
Next, we extend the classical Cayley–Hamilton theorem to the WG inverse matrix. LetA∈ℂn,n with IndA=k. From Lemma 1 and Lemma 2, it can be obtained(77)detsIn−UT−1T−2S00U∗=sn−rkAkrksIrkAk−T−1.
The characteristic polynomial ofT−1 is(78)pT−1s=detsIrkAk−T−1=srkAk+bn−1srkAk−1+⋯+bn−rkAk+1s+bn−rkAk.
It is given by the classical Cayley–Hamilton theorem(79)pT−1T−1=T−1rkAk+bn−1T−1rkAk−1+⋯+bn−rkAk+1T−1+bn−rkAkIrkAk=0.
Postmultiplying the above equation byTrkAk,(80)IrkAk+bn−1T+⋯+bn−rkAk+1TrkAk−1+bn−rkAkTrkAk=0.
By using (77) and (78), we have Theorem 5.Theorem 5.
LetA∈ℂn,n with IndA=k, then the characteristic polynomial of AⓌ is(81)ps=detsIn−A=sn+bn−1sn−1+⋯+bn−rkAksn−rkAk,AⓌn+bn−1AⓌn−1+⋯+bn−rkAkAⓌn−rkAk=0,where bn−1,…,bn−rkAk is as in (78).Example 2.
Let(82)A=1010010100010000,then IndA=2,rkA2=2, and(83)T=1001,T−1=1001,AⓌ=1010010100000000.
Furthermore,(84)pT−1s=s2−2s+1,pAⓌs=s2s2−2s+1=s4−2s3+s2,10100101000000004−210100101000000003+10100101000000002=0.


## 4. Conclusion

In this paper, we give some new characterizations of the WG matrix. Some relations between various well-known generalized inverses and the WG matrix are proved, and 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].



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