In this paper, we use the following notations.The symbol Cm,n is the set of m×n matrices with complex entries, and rk(A)represents the rank of A∈Cm,n.Let A∈Cn,n, and then the smallest non-negative integer k, which satisfies rk(Ak+1)=rk(Ak), is called the index of A and is denoted as Ind(A).The Moore-Penrose inverse of A∈Cm,n is defined as the unique matrix X∈Cn,m satisfying the equations AXA=A, XAX=X, (AX)*=AX, (XA)*=XA, and is denoted as X=A+.The Drazin inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAk+1=Ak, XAX=X, AX=XA and is usually denoted as X=AD (see Ref.[1]).The core-EP inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAk+1=Ak, XAX=X and (AX)*=AX, and is denoted as X=A⨁[2].The DMP inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAX=X, XA=ADA and AmX=AkA+, and is denoted as X=Ad,+[3].More details of the Drazin, core-EP, DMP inverses can be seen in Refs.[4-8].

The Cayley-Hamilton theorem has many applications in nonlinear time-varying systems, electric circuits, etc.The classical Cayley-Hamilton theorem was extended to the fractional continuous-time and discrete-time linear systems[9], nonlinear time-varying systems with square and rectangular systems[10], the Drazin inverse matrix and standard inverse matrix[11], etc.More details about the Cayley-Hamilton theorem and its applications can be read in Refs.[9-13].Therefore, it is very interesting to investigate the Cayley-Hamilton theorem for the core-EP inverse matrix and DMP inverse matrix.In this paper, our main tools are core-EP decomposition and generalized inverses.

1　Preliminaries

In this section, we present some preliminary results.

Theorem 1[14, Cayley-Hamilton theorem]　Let pA(s)=det(sEn-A)be the characteristic polynomial of X∈Cn,n.Then pA(A)=0.

Theorem 2[14]　Let A∈Cn,n is singular, i.e.det(A)=0, and the characteristic polynomial of A be

pA(s)=det(sEn-A)=sn+an-1sn-1+…+a1s

(1)

Then

fA(AD)=a1(AD)n+a2(AD)n-1+…+an-1(AD)2+AD=0

(2)

Lemma 1[15, core-nilpotent decomposition]　Let A∈Cn,n be with Ind(A)=k.Then A can be written as the sum of matrices and i.e. where is nilpotent, and Here one or both of and can be null.Furthermore, there is a nonsingular matrix P such that

(3)

where Σ∈Crk(A),rk(A) is non-singular, and is nilpotent and

Lemma 2[15, core-EP decomposition]　Let A∈Cn,n be with Ind(A)=k.Then A can be written as the sum of matrices A1 and A2, i.e.A=A1+A2 where Ind(A1)≤1, A2 is nilpotent, and A2=A2A1=0.Here one or both of A1 and A2 can be null.Furthermore, there is a unitary matrix U such that

(4)

where T∈Crk(A),rk(A) is non-singular, N is nilpotent and Nk=0.

Lemma 3　Let the core-EP decomposition of A∈Cn,n be as in Lemma 2.Then the core-EP inverse of A is

A⨁

(5)

Let the core-EP decomposition of A be as in (4).Then

(6)

and

(A⨁

(7)

where It is easy to confirm that Φj=Tj-kΦk, where j≥k.

2　Main Results

In this section the classical Cayley-Hamilton theorem will be extended to the core-EP inverse matrix and DMP inverse matrix.By assumption, matrix A is singular, i.e.det(A)=0.

Lemma 4　Let the characteristic polynomial of A∈Cn,nbe as in Eq.(1).Then

fA(A⨁)=a1(A⨁)n+a2(A⨁)n-1+…+

an-1(A⨁)2+A⨁=0

(8)

Proof　Using (1)and Theorem 1 we obtain

An+an-1An-1+…+a1A=0

(9)

It follows from Lemma 2 and (6)that

(10)

Post-multiplying (10)by

we have

(11)

Therefore, by applying (7), we obtain (8).

Example 1　Let

Then Ind (A)=2, the Drazin AD is

and the core-EP inverse A⊕ is

A⊕

The characteristic polynomial of A is

　　　s4-2s3+s2+0s

From the classical Cayley-Hamilton theorem, we have A4-2A3+A2=0.By applying Lemma 4, we obtain (A⊕)3-2(A⊕)2+A⊕=0.

Note that, if the characteristic polynomials of A⨁ and AD is

pA⨁(s)=det(sEn-A⨁)=sn+bn-1sn-1+…+b1s

(12)

pAD(s)=det(sEn-AD)=sn+cn-1sn-1+…+c1s

(13)

respectively, we cannot obtain

pA⨁(s)=b1An+b2An-1+…+bn-1A2+A

(14)

pAD(s)=c1An+c2An-1+…+cn-1A2+A

(15)

Example 2　Let

It is easy to confirm that the core-EP inverse A⊕ is

A⊕

Then

but

fAD(s)=fA⊕(s)=0A2+A=A≠0

Theorem 3　Let A∈Cn,n and Ind(A)=k.Then the characteristic polynomial of A⨁∈Cn,n is

pA⨁(s)=sn+bn-1sn-1+…+bn-rk(Ak)sn-rk(Ak)

(16)

Furthermore,

bn-rk(Ak)An+…+bn-1An-rk(Ak)+1+An-rk(Ak)=0

(17)

Proof　Let the core-EP decomposition of A, i.e.A=A1+A2, be as in Lemma 2.Then

sn-rk(Ak)det(sErk(Ak)-T-1)

Therefore, we obtain (16).Using (16)and Theorem 1, we obtain

(A⨁)n+bn-1(A⨁)n-1+…+bn-rk(Ak)(A⨁)n-rk(Ak)=0

(18)

that is,

(19)

Post-multiplying (19)by

we have (17).

Theorem 4　Let A∈Cn,n, Ind(A)≤1 and the characteristic polynomial of A⨁∈Cn,n be as in (12).Then

fA⨁(A)=b1An+b2An-1+…+bn-1A2+A=0

(20)

Theorem 5　Let A∈Cn,n and Ind(A)=k.Then the characteristic polynomial of AD∈Cn,n is

pAD(s)=det(sEn-AD)=sn+cn-1sn-1+…+cn-rk(Ak)sn-rk(Ak)

(21)

Furthermore,

cn-rk(Ak)An+…+cn-1An-rk(Ak)+1+An-rk(Ak)=0

(22)

Proof　Let the core-nilpotent decomposition of A, be as in Lemma 1.Then

sn-rk(Ak)det(sErk(Ak)-Σ-1)

Therefore, we obtain (21).Using (21)and Theorem 1 we obtain

(AD)n+bn-1(AD)n-1+…+bn-rk(Ak)(AD)n-rk(Ak)=0

that is,

Post-multiplying the above equation by

Therefore, we obtain (22).

Theorem 6　Let A∈Cn,n and Ind(A)≤1 and the characteristic polynomial of AD∈Cn,n be as in (13).Then

fAD(A)=c1An+c2An-1+…+cn-1A2+A=0

(23)

Let A∈Cn,n and Ind(A)=k.Then the DMP inverse of A is Ad,+=ADAA+[3].Since (Ad,+)2=ADAA+ADAA+=ADAA+AADA+=(AD)2AA+, we obtain

(Ad,+)p=(AD)pAA+

(24)

where p is a positive integer.

Theorem 7　Let the characteristic polynomial of A∈Cn,n be as in (1).Then

fA(Ad,+)=a1(Ad,+)n+a2(Ad,+)n-1+…+

an-1(Ad,+)2+Ad,+=0

(25)

Proof　By applying (1)and Theorem 2, we obtain

a1(AD)nAA++a2(AD)n-1AA++…+

an-1(AD)2AA++ADAA+=0

From (24), we can obtain (25).

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