A new generalized inverse of matrices from core-EP decomposition
aa r X i v : . [ m a t h . R A ] J u l A new generalized inverse of matrices from core-EP decomposition
Kezheng Zuo , Yu Li , Gaojun Luo Tuesday 7 th July, 2020
Abstract
A new generalized inverse for a square matrix H ∈ C n × n , called CCE-inverse, is establishedby the core-EP decomposition and Moore-Penrose inverse H † . We propose some characterizationsof the CCE-inverse. Furthermore, two canonical forms of the CCE-inverse are presented. At last,we introduce the definitions of CCE-matrices and k -CCE matrices, and prove that CCE-matricesare the same as i -EP matrices studied by Wang and Liu in [The weak group matrix, AequationesMathematicae, 93(6): 1261-1273, 2019]. AMS classification : 15A09; 15A03.
Keywords : CCE-inverse; Moore-Penrose inverse; Core-EP decomposition; Core-EP inverse; EP-matrix
Generalized inverses of matrices are closely associated with orthonormalization, linear equations,singular values, least squares solutions and various matrix factorizations. Over the past few decades,there has been an increasing interest in the study of generalized inverses due to their wide utilizationin many fields such as statistics [23], neural network [16], compressed sensing [6] and so on.Let C m × n stand for the set of m × n complex matrices. Denote the range space, null space,conjugate transpose and rank of H ∈ C m × n by R ( H ), N ( H ), H ∗ and r ( H ), respectively. Moreover I n will be the identity matrix of order n . For a matrix H ∈ C n × n , the index of H is said to be Department of Mathematics, Hubei Normal University, Hubei, Huangshi, China. (Email: [email protected]) Department of Mathematics, Hubei Normal University, Hubei, Huangshi, China. (Email: [email protected]) Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China. (Email:[email protected])This work was supported by NSFC of China (Grant No. 11961076). k > r ( H k ) = r ( H k +1 ) and written Ind( H ). Let C n × nk be the setconsisting of n × n complex matrices with index k . For a matrix H ∈ C m × n , if there exists a uniquematrix X ∈ C n × m satisfying(1) HXH = H, (2) XHX = X, (3) ( HX ) ∗ = HX, (4) ( XH ) ∗ = XH, then X = H † is termed Moore-Penrose (MP for short) inverse of H [22]. Furthermore, a matrix X ∈ C n × m satisfying the condition (1) is called a g -inverse of H . A matrix X ∈ C n × m is said to bean outer inverse of H if XHX = X and is denoted by H (2) . If there exists a matrix X ∈ C n × m thatsatisfies the conditions (1) and (2), then it is referred to as a reflexive g-inverse of H . A matrix X satisfying X = XHX , R ( H ) = T and N ( H ) = S is denoted by A (2) T , S . Basing on the MP-inverse,one can obtain the orthogonal projection onto R ( H ), which is represented by P H = HH † .The Drazin inverse H D ∈ C n × n of H ∈ C n × nk is the unique solution of the following equations [8]: HH D H = H, HH D = H D H, H D H k +1 = H k . Especially, the Drazin inverse of H is reduced to the group inverse of H which is denoted by H ifInd( H ) = 1. In general, nonsingular matrices and matrices with index 1 have nice algebraic structure.Assume that C CM n is the set consisting of matrices with order n and index less than or equal to 1.For any matrix H ∈ C CM n , the authors [3] established a new generalized inverse, called core inverse,which is the unique solution of HX = P H , R ( X ) ⊆ R ( H ) . Denote by H (cid:13) the core inverse of H ∈ C CM n . The core inverse has been an important concept inthe study of the generalized inverses (see [3, 14, 15, 24]). Moreover, several generalizations of thecore inverse, i.e., core-EP inverse, DMP-inverse, BT-inverse, ( B, C )-inverse, weak group inverse andCMP-inverse, were provided by the unique solution of several equations as follows.For a matrix H ∈ C n × nk , the unique solution H †(cid:13) ∈ C n × n of the equations H †(cid:13) HH †(cid:13) = H †(cid:13) , R ( H †(cid:13) ) = R (( H †(cid:13) ) ∗ ) = R ( H k ) , is called the core-EP inverse of H ([see [11, 21, 26, 33]).The DMP-inverse [20, 34] H D, † of H ∈ C n × nk is defined as H D, † HH D, † = H D, † , H D, † H = H D H, H k H D, † = H k H † . H D, † = H D HH † . Also, the dual DMP-inverse of H is definedto be the matrix H † ,D = H † HH D [20].Baksalary et al. [2] introduced the BT-inverse H ♦ of H ∈ C n × n , which is defined by H ♦ = ( H H † ) † = ( HP H ) † . The (
B, C )-inverse [4, 7] of H ∈ C m × n , denoted by H ( B,C ) , is the unique matrix X ∈ C n × m suchthat CHX = C, XHB = B, R ( X ) = R ( B ) , N ( X ) = N ( C ) , where B, C ∈ C n × m .In 2018, Wang and Chen [27] defined the weak group inverse H w (cid:13) of H ∈ C n × nk by( H w (cid:13) ) = H w (cid:13) , HH w (cid:13) = H †(cid:13) H. Very recently, Mehdipour and Salem [19] proposed a new generalized inverse, termed the CMP-inverse of H ∈ C n × nk , written as H C, † , which is defined by H C, † HH C, † = H C, † , HH C, † H = e H , HH C, † = e H H † , H C, † H = H † e H , where the matrix e H is the core part of core-nilpotent decomposition of H (cid:0) in fact, e H = HH D H ,see [19, 31] (cid:1) .Note that the CMP-inverse is derived from the core-nilpotent decomposition and MP-inverse.Wang [26] proposed a new decomposition of H ∈ C n × nk , which is referred to as core-EP (CEP forsimplicity) decomposition. It is natural to consider the CEP-decomposition in the definition of theCMP-inverse. As a result, we get a new generalized inverse called the CCE-inverse. In this work,we introduce the CCE-inverse H C, †(cid:13) for square matrices of an arbitrary index using its core part H from the CEP-decomposition of H and its MP-inverse H † . Using the CEP-decomposition of H ,we derive some characterizations of the CCE-inverse of H . Meanwhile, we introduce CCE-matricesand k -CCE matrices and show that these two kinds of matrices are consistent. Finally, we study therelationships between CCE-matrices and k -CCE matrices with some special matrix classes.The rest of the material is organized as follows. In Section 2, a new generalized inverse isestablished and we study its characterizations. In Section 3, two canonical forms of CCE-inverse andtheir applications are presented. In Section 4, we investigate CCE-matrices and k -CCE matrices.3 A new generalized inverse on core-EP decomposition
In this section, we propose a new generalized inverse of H by the CEP-decomposition and MP-inverse. We begin with the CEP-decomposition.Wang [26] introduced the CEP-decomposition of H ∈ C n × nk , which says that a matrix H ∈ C n × nk can be represented in the following form: H = H + H = U T P Q U ∗ , H = U T P U ∗ , H = U Q U ∗ , (2.1)where T is nonsingular with r ( T ) = r ( H k ) and Q is nilpotent with index k and U ∈ C n × n isunitary. The expression of H provided by (2 .
1) is unique and satisfies Ind( H ) ≤ H k = 0 and H ∗ H = H H = 0 [26, Theorem 2.1]. In (2 . H and H are termed the core part and nilpotentpart of H , respectively. In addition, it is easy to verify that H = HH †(cid:13) H . For simplicity, wealways assume that the matrix H ∈ C n × nk has the CEP-decomposition H = H + H throughout thissection. Definition 2.1.
Suppose that H ∈ C n × nk . Then the CCE-inverse of H is defined by H C, †(cid:13) = H † H H † = H † HH †(cid:13) HH † = Q H H †(cid:13) P H , where P H = HH † and Q H = H † H . Theorem 2.2.
Assume that H ∈ C n × nk . Then X = H C, †(cid:13) is the unique solution of the followingthree equations XHX = X, HX = H H † , XH = H † H . (2.2) Proof.
Obviously, H C, †(cid:13) = H † H H † is a solution of this system. For uniqueness, suppose that X satisfies (2 . X = XHX = X ( H H † )= X ( HH †(cid:13) H ) H † = ( H † H )( H †(cid:13) HH † )= H † H H † . Hence, X = H C, †(cid:13) is the unique matrix such that XHX = X, HX = H H † , XH = H † H .In the following, we use a concrete sample to illustrate the difference between the CCE-inverseand other generalized inverses. 4 xample 2.3. Let H = I I N , where N = . It is easy to check that Ind ( H ) = 3 .By a direct computation, we obtain the generalized inverses as follows: H † = H − N † I − H N † , H D = I H ,H D, † = H D HH † = I H , H † ,D = H † HH D = H H I − H H − H ,H ⋄ = ( H H † ) † = H − H I − H H , H †(cid:13) = I
00 0 ,H C, † = Q H H D P H = H H I − H H − H , H w (cid:13) = I I . It is easy to see that the CCE-inverse H C, †(cid:13) is H C, †(cid:13) = H I − H , where H = , H = , H = , H =
12 12 12 ,H =
00 0 1 , H = , H =
12 12
00 1 00 0 0 and N † = . According to Example 2.3, the CCE-inverse is indeed a new generalized inverse. Now, we showthat the CCE-inverse H C, †(cid:13) of H is an outer inverse of H (i.e., H C, †(cid:13) HH C, †(cid:13) = H C, †(cid:13) ) and areflexive g-inverse of H . Theorem 2.4.
Let H ∈ C n × nk with CEP-decomposition H = H + H defined by (2 . . Then(a) H C, †(cid:13) is an outer inverse of H ;(b) H C, †(cid:13) is a reflexive g-inverse of H . roof. (a). This is evident.(b). To prove the result, we need to show H H C, †(cid:13) H = H , H C, †(cid:13) H H C, †(cid:13) = H C, †(cid:13) . Using the properties of H † and H †(cid:13) , it is easy to prove both of two equations above. Hence, theresult holds. Theorem 2.5.
Let H ∈ C n × nk . Then the following assertions are equivalent:(a) H ∈ C CM n ;(b) H C, †(cid:13) ∈ H { } ( i.e., HH C, †(cid:13) H = H ) ;(c) H C, †(cid:13) = H † .Proof. “( a ) ⇔ ( b )”. It can be easily seen that A C, †(cid:13) ∈ H { } if and only if H = 0 which is equivalentto N = 0, i.e. r ( H ) = r ( H ).“( b ) ⇔ ( c )”. Premultiplying and postmultiplying H C, †(cid:13) = H † by H , we obtain H C, †(cid:13) ∈ H { } .Premultiplying and postmultiplying HH C, †(cid:13) H = H by H † we have H C, †(cid:13) = H † . The the desiredresult follows. Theorem 2.6.
Let H ∈ C n × nk . Then H C, †(cid:13) = H † P H k = H † ,D P H k = H C, † P H k .Proof. Since HH †(cid:13) = H k ( H k ) † (see Corollary 3.4 of [26]), we have H C, †(cid:13) = H † ( HH †(cid:13) ) HH † = H † P H k P H = H † P H k . Meanwhile, we have H † ,D P H k = H † HH D H k ( H k ) † = H † H k ( H k ) † = H † P H k = H C, †(cid:13) . Observe that H C, † P H k = H † ( HH D H ) H † P H k = H † HH D P H k = H † ,D P H k = H C, †(cid:13) . Therefore, H C, †(cid:13) = H † P H k = H † ,D P H k = H C, † P H k . Corollary 2.7.
Let H ∈ C n × nk . Then r ( H C, †(cid:13) ) = r ( H k ) , R ( H C, †(cid:13) ) = R ( H † H k ) and N ( H C, †(cid:13) ) = N (( H k ) ∗ ) . Proof.
By Theorem 2.6, we deduce that R ( H C, †(cid:13) ) = R ( H † P H k ) = H † R ( P H k ) = H † R ( H k ) = R ( H † H k ) , N ( H C, †(cid:13) ) = N ( H † P H k ) = N ( P H k ) = N (( H k ) ∗ ) . Due to R ( H C, †(cid:13) ) = R ( H † H k ), we get that r ( H C, †(cid:13) ) = r ( H † H k ) = r ( H k ) . heorem 2.8. Let H ∈ C n × nk . Then(a) HH C, †(cid:13) is an orthogonal projection onto R ( H k ) , i.e., HH C, †(cid:13) = P H k ;(b) H C, †(cid:13) H is a projector onto R ( H † H k ) along N (( H k ) † H ) , i.e., H †(cid:13) H = P R ( H † H k ) , N (( H k ) † H ) .Proof. (a). By Theorem 2.6, we deduce that HH C, †(cid:13) = HH † P H k = P H k .(b). Since H C, †(cid:13) is an outer inverse of H , H C, †(cid:13) H is a projector. It follows from R ( H C, †(cid:13) H ) = R ( H C, †(cid:13) ) = R ( H † H k ) , and N ( H C, †(cid:13) H ) = N ( H † P H k H ) = N ( P H k H ) = N (( H k ) † H ) that H C, †(cid:13) H = P R ( H † H k ) , N (( H k ) † H ) . Theorem 2.9.
Let H ∈ C n × nk . Then the following assertions are equivalent:(a) X = H C, †(cid:13) ;(b) R ( X ∗ ) = R ( H k ) and XH = H † H ;(c) R ( X ∗ ) = R ( H k ) and XH k = H † H k ;(d) R ( X ∗ ) = R ( H k ) and XH D = H † H D ;(e) R ( X ∗ ) = R ( H k ) and XHH D = H † ,D .Proof. “( a ) ⇒ ( b )”. The result is evident by Theorem 2.2 and Corollary 2.7.“( b ) ⇒ ( c )”. Note that H H k − = P H k H k = H k . Postmultiplying XH = H † H by H k − , weobtain XH k = H † H k . “( c ) ⇒ ( d )”. Postmultiplying XH k = H † H k by ( H D ) k +1 , we get that XH D = H † H D .“( d ) ⇒ ( e )”. Postmultiplying XH D = H † H D by H , we have XHH D = H † ,D .“( e ) ⇒ ( a )”. Postmultiplying XHH D = H † ,D by H k , we get that XH k = H † H k . Postmultiplying XH k = H † H k by ( H k ) † , we have XP H k = H † P H k . It follows from R ( X ∗ ) = R ( H k ) that X = XP H k .Thanks to Theorem 2.6, we obtain X = H † P H k = H C, †(cid:13) . Theorem 2.10.
Let H ∈ C n × nk . Then the following assertions are equivalent:(a) X = H C, †(cid:13) ; (b) XH X = X, H X = H A † and XH = H † H ;(c) r ( X ) = r ( H k ) , H X = H H † and XH = H † H ; d) r ( X ) = r ( H k ) , HX = H H † and XH = H † H ;(e) XP H k = X , XH = H † H .Proof. “( a ) ⇒ ( b )”. We need to check the following three equations: H C, †(cid:13) H H C, †(cid:13) = H C, †(cid:13) , H H C, †(cid:13) = ( HH †(cid:13) H ) H † , H C, †(cid:13) H = H † H . “( b ) ⇒ ( c )”. According to X = XH X , we derive that r ( X ) = r ( H X ) = r ( H H † ) = r ( HH †(cid:13) HH † ) = r ( P H k P H ) = r ( P H k ) = r ( H k ) . “( c ) ⇒ ( a )”. By H X = H H † = P H k , we can obtain that R ( H k ) = R ( P H k ) = R (( P H k ) ∗ ) = R (( H X ) ∗ ) ⊆ R ( X ∗ ) . Noting that r ( X ) = r ( H k ) we get that R ( X ∗ ) = R ( H k ). Postmultiplying XH = H † H by H †(cid:13) and using H H †(cid:13) = P H k , we get that XP H k = H † P H k . Postmultiplying XP H k = H † P H k by H k , we have XH k = H † H k . Thus, due to (c) of Theorem 2.9, X = H C, †(cid:13) .“( a ) ⇒ ( d )”. By Theorem 2.2 and Corollary 2.7, the desired result follows.“( d ) ⇒ ( a )”. The proof is analogous to that of ( c ) ⇒ ( a ).“( a ) ⇒ ( e )”. According to Theorem 2.2 and Theorem 2.6, the desired result follows.“( e ) ⇒ ( a )”. Note that X = XP H k = ( XH )( H k − ( H k ) † ) = H † ( HH †(cid:13) H )( H k − ( H k ) † ) = H † P H k P H k = H † P H k . Then X = H C, †(cid:13) .It is widely known that the generalized inverse can be expressed as a special kind of outer inversealong with prescribed range and null space. Therefore, we will prove that the same property holdsin the case of CCE-inverse. Theorem 2.11.
Let H ∈ C n × nk . Then H C, †(cid:13) = H (2) R ( H † H k ) , N (( H k ) ∗ ) . Proof.
From the definition of CCE-inverse, we know that H C, †(cid:13) is an outer inverse of H . By Corollary2.7, we have R ( H C, †(cid:13) ) = R ( H † H k ) , N ( H C, †(cid:13) ) = N (( H k ) ∗ ) . So, we obtain this result.Next, we propose a connection between the (
B, C ) − inverse and CCE-inverse, which shows thatthe CCE-inverse of a matrix H ∈ C n × nk is the ( H † H k , ( H k ) ∗ ) − inverse of H .8 heorem 2.12. Let H ∈ C n × nk . Then H C, †(cid:13) = H ( H † H k , ( H k ) ∗ ) . Proof.
Employing Corollary 2.7, we can obtain R ( H C, †(cid:13) ) = R ( H † H k ) , N ( H C, †(cid:13) ) = N (( H k ) ∗ ) . Observe that H C, †(cid:13) H ( H † H k ) = H † P H k H k = H † H k , ( H k ) ∗ HH C, †(cid:13) = ( H k ) ∗ H ( H † HH †(cid:13) HH † )= ( H k ) ∗ ( HH †(cid:13) )( HH † ) = ( H k ) ∗ P H k = ( P H k H k ) ∗ = ( H k ) ∗ . Thus, we obtain H C, †(cid:13) = H ( H † P Hk , ( H k ) ∗ ) . In this section, we present two canonical forms of CCE-inverse by matrix decompositions, whichis used to study the properties of CCE-inverse of H . Before that, we need to do some preparations.For convenience, we adopt the following notation in the sequel. • C PI m,n = { H | H ∈ C m × n , HH ∗ H = H } denotes the set of comprising partial isometries; • Let C P n = { H | H ∈ C n × n , H = H } be the set of projectors; • C OP n = { H | H ∈ C n × n , H = H = H ∗ } is the set of orthogonal projectors; • C EP n = { H | H ∈ C n × n , HH † = H † H } is the set of EP matrices; • C k, † n = C k − EP = { H | H ∈ C n × nk , H k H † = H † H k } denotes the set of k -EP matrices; • C i − EP n = { H | H ∈ C n × nk , H k ( H k ) † = ( H k ) † H k } denotes the set of i -EP matrices; • C k, †(cid:13) n = { H | H ∈ C n × nk , H k H †(cid:13) = H †(cid:13) H k } is the set of k -Core EP matrices; • Let C k,D † n = { H | H ∈ C n × nk , H k H D, † = H D, † H k } be the set of k -DMP matrices; • Let C k, † Dn = { H | H ∈ C n × nk , H k H † ,D = H † ,D H k } be the set of dual k -DMP matrices; • C k,C † n = { H | H ∈ C n × nk , H k H C, † = H C, † H k } denotes the set of k -CMP matrices.9 .1 The first canonical form derived from the Hartwig-Spindelbock decomposi-tion On the basis of Corollary 6 in [13], every H ∈ C n × n with rank r can be expressed by H = U Σ M Σ N U ∗ , (3.1)where U ∈ C n × n is unitary, Σ = diag ( σ , σ , . . . , σ r ) is a diagonal matrix whose diagonal entries σ i > i = 1 , , · · · , r ) are singular values of H and M ∈ C r × r , N ∈ C r × ( n − r ) such that M M ∗ + N N ∗ = I r . (3.2)According to the above decomposition, a straightforward computation shows that H † = U M ∗ Σ − N ∗ Σ − U ∗ , P H = HH † = U I r
00 0 U ∗ . (3.3)It is well-known that (see [3, 11]): H = U (Σ M ) − M − Σ − M − N U ∗ , H (cid:13) = U (Σ M ) −
00 0 U ∗ ,H †(cid:13) = U (Σ M ) †(cid:13)
00 0 U ∗ . (3.4)Using the decomposition of (3.1), we have the following properties. Lemma 3.1. [1]
Assume that H ∈ C n × n with rank r is decomposed by (3 . . Then,(a) H ∈ C CM n ⇔ M is nonsingular;(b) H ∈ C P n ⇔ Σ M = I r ;(c) H ∈ C OP n ⇔ N = 0 , Σ = M = I r ;(d) H ∈ C PI n,n ⇔ Σ = I r ;(e) H ∈ C EP n ⇔ N = 0 . Employing (3 . .
3) and (3 . heorem 3.2. Assume that H ∈ C n × n has the form of (3 . and r ( H ) = r . Then H C, †(cid:13) = U M ∗ M (Σ M ) †(cid:13) N ∗ M (Σ M ) †(cid:13) U ∗ . (3.5) Proof.
By using (3 . .
3) and (3 . H C, †(cid:13) = H † H H † = U M ∗ Σ − N ∗ Σ − Σ M (Σ M ) †(cid:13)
00 0 I r
00 0 U ∗ = U M ∗ M (Σ M ) †(cid:13) N ∗ M (Σ M ) †(cid:13) U ∗ . By the expression of H C, †(cid:13) in Theorem 3.2, we gather the following results. Theorem 3.3.
Let H ∈ C n × n , r ( H ) = r . Then(a) H C, †(cid:13) = 0 ⇔ H is a nilpotent matrix;(b) H C, †(cid:13) = P H ⇔ H ∈ C OP n ;(c) H C, †(cid:13) = H ⇔ H ∈ C EP n and H = H ;(d) H C, †(cid:13) = H ⇔ H ∈ C EP n ;(e) H C, †(cid:13) = H (cid:13) ⇔ H ∈ C EP n .Proof. (a). By using the Theorem 3.2, we get that H C, †(cid:13) = 0 ⇔ M ∗ M (Σ M ) †(cid:13) = 0 and N ∗ M (Σ M ) †(cid:13) = 0 . Combining them with
M M ∗ + N N ∗ = I r leads to M (Σ M ) †(cid:13) = 0 . Then (Σ M ) †(cid:13) = 0 , which meansthat H †(cid:13) = 0 . Therefore, H is a nilpotent matrix.Conversely, it is evident.(b). “ ⇐ ”. If H ∈ C OP n , then H = U I r
00 0 U ∗ , Thus, it is easy to see that H C, †(cid:13) = P H . 11 ⇒ ”. Assume that H C, †(cid:13) = P H . By Theorem 3.2 and (3 . U M ∗ M (Σ M ) †(cid:13) N ∗ M (Σ M ) †(cid:13) U ∗ = U I r
00 0 U ∗ . Thus, M ∗ M (Σ M ) †(cid:13) = I r , N ∗ M (Σ M ) †(cid:13) = 0 . (3.6)Due to M M ∗ + N N ∗ = I r , it follows from (3 .
6) that N = 0 , Σ M = I r . Hence, H = U I r
00 0 U ∗ ∈ C OP n . (c). “ ⇐ ”. If H ∈ C EP n and H = H , by (3 .
1) and Lemma 3.1, we get that H = U Σ M
00 0 U ∗ , (Σ M ) = I r . Thus, H C, †(cid:13) = H is evident.“ ⇒ ”. Let H C, †(cid:13) = H . by (3 .
1) and Theorem 3.2, we derive that U M ∗ M (Σ M ) †(cid:13) N ∗ M (Σ M ) †(cid:13) U ∗ = U Σ M Σ N U ∗ , which implies that N = 0 , M ∗ M = I r , (Σ M ) †(cid:13) = (Σ M ) − = Σ M. Thus, we have H = H and H ∈ C EP n . (d). Using (3 . .
4) and Theorem 3.2, we show that H C, †(cid:13) = H if and only if N = 0.Therefore, H C, †(cid:13) = H ⇔ H ∈ C EP n . (e). Using the same argument as in the proof of (d), the result can be easily carried out. In Section 2, we have introduced the CEP-decomposition. In fact, several generalized inverseshave the following expressions by utilizing the CEP-decomposition.
Lemma 3.4. [26, Theorem 3.2]
Let H ∈ C n × nk be decomposed by (2 . . Then H †(cid:13) = H (cid:13) . Further-more H †(cid:13) = U T −
00 0 U ∗ . (3.7)12 emma 3.5. [12, 30] Let H ∈ C n × nk be decomposed by (2 . . Then H D = U T − ( T k +1 ) − e T U ∗ , (3.8) where e T = k − P j =0 T j P Q k − − j . Furthermore, e T = 0 if and only if P = 0 . Lemma 3.6. [9]
Let H ∈ C n × nk have the form of (2 . . Then H † = U T ∗ Λ − T ∗ Λ P Q † ( I n − t − Q † Q ) P ∗ Λ Q † − ( I n − t − Q † Q ) P ∗ Λ P Q † U ∗ , (3.9) where Λ = (
T T ∗ + P P ∗ − P Q † QP ∗ ) − , t = r ( H k ) . Moreover, HH † = U I t QQ † U ∗ , (3.10) H † H = U T ∗ Λ T T ∗ Λ P ( I n − t − Q † Q )( I n − t − Q † Q ) P ∗ Λ T Q † Q + ( I n − t − Q † Q ) P ∗ Λ P ( I n − t − Q † Q ) U ∗ . (3.11)Utilizing the above expressions of generalized inverses, several characterizations of the sets con-sisting of special matrices were introduced as follows. Lemma 3.7. [12]
Let H ∈ C n × nk be given by (2 . . Then(a) H ∈ C i − EPn ⇔ H ∈ C k, †(cid:13) n ⇔ P = 0 ;(b) H ∈ C k − EPn ⇔ H ∈ C k,C † n ⇔ P = P Q † Q and e T = e T QQ † , where e T = k − P j =0 T j P Q k − j ;(c) H ∈ C k,D † n ⇔ e T = e T QQ † ;(d) H ∈ C k, † Dn ⇔ P = P Q † Q ;(e) H ∈ C k, w (cid:13) n ⇔ P Q = 0 . By Theorem 2.6 and the CEP-decomposition, we derive another canonical form of the CCE-inverse.
Theorem 3.8.
Let H ∈ C n × nk be expressed as in (2 . . Then H C, †(cid:13) = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ , (3.12)13here Λ = ( T T ∗ + P P ∗ − P Q † QP ∗ ) − , t = r ( H k ) . Proof.
By Theorem 2.6 and (3 . H C, †(cid:13) = H † P H k = U T ∗ Λ − T ∗ Λ P Q † ( I n − t − Q † Q ) P ∗ Λ Q † − ( I n − t − Q † Q ) P ∗ Λ P Q † I t
00 0 U ∗ = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ , where Λ = ( T T ∗ + P P ∗ − P Q † QP ∗ ) − , t = r ( H k ).Now, some properties of H C, †(cid:13) are deduced by Theorems 3.8 in the following two theorems. Theorem 3.9.
Let H ∈ C n × nk have the form of (2 . . Then(a) H C, †(cid:13) = H D ⇔ P = 0 ⇔ H ∈ C i − EP n (or equivalently H k ∈ C EP n );(b) H C, †(cid:13) = H † ,D ⇔ H ∈ C i − EP n ;(c) H C, †(cid:13) = H ∗ ⇔ H ∈ C CM n ∩ C PI n ;(d) H C, †(cid:13) = H †(cid:13) ⇔ P = P Q † Q ⇔ H ∈ C k, † Dn (or equivalently H k H † D = H † D H k ) ;(e) H C, †(cid:13) = H D, † ⇔ H ∈ C k, w (cid:13) n ∩ C k, † D n ;(f ) H C, †(cid:13) = H C, † ⇔ P = P Q † Q ⇔ H ∈ C k, † D n .Proof. (a). By (3 .
8) and (3 . H C, †(cid:13) = H D ⇔ U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ = U T − ( T k +1 ) − e T U ∗ ⇔ T ∗ Λ = T − , e T = 0 , P = P Q † Q ⇔ P = 0 . (b). Thanks to (3 .
8) and (3 . H † D is given by H † D = H † HH D = U T ∗ Λ T ∗ Λ T − k e T ( I n − t − Q † Q ) P ∗ Λ ( I n − t − Q † Q ) P ∗ Λ T − k e T U ∗ . (3.13)Then, H † D = H C, †(cid:13) ⇔ e T = 0 ⇔ P = 0. 14c). It is easy to check that H C, †(cid:13) = H ∗ ⇔ U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ = U T ∗ P ∗ Q ∗ U ∗ ⇔ Q ∗ = 0 , T ∗ Λ = T ∗ and ( I n − t − Q † Q ) P ∗ = P ∗ ⇔ Q = 0 and T T ∗ + P P ∗ = I t ⇔ H = U T P U ∗ and HH ∗ H = H ⇔ H ∈ C CM n ∩ C PI n . (d). By (3 . .
12) and Lemma 3.7, a straightforward computation shows that H C, †(cid:13) = H †(cid:13) ⇔ T ∗ Λ = T − and ( I n − t − Q † Q ) P ∗ Λ = 0 ⇔ P = P Q † Q ⇔ H ∈ C k, † Dn . (e). It follows from (3 .
8) and (3 .
10) that H D, † = H D HH † = U T − ( T k +1 ) − e T I QQ † U ∗ = U T − ( T k +1 ) − e T QQ † U ∗ . According to (3 . H C, †(cid:13) = H D, † ⇔ T − = T ∗ Λ , ( I n − t − Q † Q ) P ∗ Λ = 0 and ( T k +1 ) − e T QQ † = 0 ⇔ P = P Q † Q and e T QQ † = 0 ⇔ P = P Q † Q and e T Q = 0 ⇔ P = P Q † Q and P Q = 0 ⇔ H ∈ C k, w (cid:13) n ∩ C k, † Dn ( by Lemma . . (f). By (3 . .
9) and (3 . H C, † = U T ∗ Λ T ∗ Λ T − k e T QQ † ( I n − t − Q † Q ) P ∗ Λ ( I n − t − Q † Q ) P ∗ Λ T − k e T QQ † U ∗ . (3.14)15herefore, H C, †(cid:13) = H C, † ⇔ e T QQ † = 0 ⇔ e T Q = 0 ⇔ P Q = 0 ⇔ H ∈ C k, w (cid:13) n (or equivalently H k H w (cid:13) = H w (cid:13) H k ). Theorem 3.10.
Let H ∈ C n × nk have the form of (2 . . Then the following assertions are equivalent:(a) H C, †(cid:13) ∈ C EP n ;(b) P = P Q † Q ;(c) H k H † ,D = H † ,D H k ;(d) H C, †(cid:13) H k +1 = H k ;(e) H † H k +1 = H k ;(f ) H C, †(cid:13) H k = H w (cid:13) H k ;(g) H C, † H k = H w (cid:13) H k ;(h) HH D = H † ,D H ;(i) HH D = H C, † H .Proof. “( a ) ⇔ ( b )”. By Theorem 3.8, we have H C, †(cid:13) = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ , where Λ = [ T T ∗ + P ( I − Q † Q ) P ∗ ] − . Therefore, H C, †(cid:13) ∈ C EP n ⇔ R ( H C, †(cid:13) ) = R ( H ( C, †(cid:13) ) ∗ ) ⇔ ( I − Q † Q ) P ∗ Λ = 0 ⇔ P = P Q † Q. “( b ) ⇔ ( c )”. The result can be proved by Theorem 3.16 of [12].“( b ) ⇔ ( d )”. Due to H = U T P Q U ∗ , we get that H k = U T k e T U ∗ , H k +1 = U T k +1 T e T U ∗ , where e T = T k − P + T k − P Q + · · · + T P Q k − + P Q k − . Thus, we have H C, †(cid:13) H k +1 = H k ⇔ T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 T k +1 T e T = T k e T ⇔ P = P Q † Q. e ) ⇔ ( d )”. One can easily verify that H C, †(cid:13) H k +1 = H k ⇔ H † P H k H k +1 = H k ⇔ H † H k +1 = H k .“( b ) ⇔ ( f )”. Thanks to H = U T P Q U ∗ , we obtain that H C, †(cid:13) = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ , H w (cid:13) = U T − T − P U ∗ (seeTheorem3 . ,H k = U T k e T U ∗ , where Λ = [ T T ∗ + P ( I − Q † Q ) P ∗ ] − , e T = T k − P + T k − P Q + · · · + T P Q k − + P Q k − . Therefore, H C, †(cid:13) H k = H w (cid:13) H k ⇔ ( I − Q † Q ) P ∗ Λ T k ⇔ P = P Q † Q. “( b ) ⇔ ( g )”. By (3 . b ) ⇔ ( f ).“( b ) ⇔ ( h )”. It follows from (3 .
8) and (3 .
13) that HH D = H † ,D H ⇔ T P Q T − ( T k +1 ) − e T = T ∗ Λ T ∗ Λ T − k e T ( I n − t − Q † Q ) P ∗ Λ ( I n − t − Q † Q ) P ∗ Λ T − k e T T P Q ⇔ I T − k e T = T ∗ Λ T T ∗ Λ P + T ∗ Λ T − k e T Q ( I n − t − Q † Q ) P ∗ Λ T ( I n − t − Q † Q ) P ∗ Λ P + ( I n − t − Q † Q ) P ∗ Λ T − k e T Q ⇔ I = T ∗ Λ TT − k e T = T ∗ Λ P + T ∗ Λ T − k e T Q I n − t − Q † Q ) P ∗ Λ T I n − t − Q † Q ) P ∗ Λ P + ( I n − t − Q † Q ) P ∗ Λ T − k e T Q. ⇔ P = P Q † Q. “( h ) ⇔ ( i )”. As H C, † H = H † ,D H , then HH D = H C, † H ⇔ HH D = H † ,D H . In this section, we generalize the definitions of the EP-matrix and p -EP matrix [18] to the CCE-inverse matrix and k-CCE matrix, respectively. Furthermore, we investigate their characterizationsby applying the CEP-decomposition. We start with the concept of CCE-inverse matrix. Definition 4.1.
Let H ∈ C n × nk . Then H is called a CCE-inverse matrix if HH C, †(cid:13) = H C, †(cid:13) H . heorem 4.2. Let H ∈ C n × n have the form of (2 . . Then the following assertions are equivalent:(a) HH C, †(cid:13) = H C, †(cid:13) H ;(b) H H C, †(cid:13) = H C, †(cid:13) H ;(c) H H † = H † H ;(d) P = 0 (or equivalently H k ∈ C EP n ).Proof. Since HH † H = H and H †(cid:13) HH †(cid:13) = H †(cid:13) , we get that ( a ) ⇔ ( b ) ⇔ ( c ).“( c ) ⇒ ( d )”. Suppose that H H † = H † H . Then we have HH †(cid:13) HH † = H † HH †(cid:13) H . Notingthat HH †(cid:13) HH † = H † HH †(cid:13) H if and only if P H k = H † P H k H , we deduce that U I t
00 0 U ∗ = U T ∗ Λ − T ∗ Λ P Q † ( I n − t − Q † Q ) P ∗ Λ Q † − ( I − Q † Q ) P ∗ Λ P Q † I t
00 0
T P Q U ∗ . Thus, I t
00 0 = T ∗ Λ T T ∗ Λ P ( I n − t − Q † Q ) P ∗ Λ T ( I n − t − Q † Q ) P ∗ Λ P . By the above equation, we have P = 0 as T ∗ Λ is nonsingular.“( d ) ⇒ ( c )”. Assume that H = U T Q U ∗ , which is from the CEP-decomposition (2 . H = HH †(cid:13) H = U T
00 0 U ∗ , H † = U T − Q † U ∗ , which implies that H H † = H † H .Define the matrix operation as [ E, F ] = EF − F E . Next, using the fact that H is the CCE-inversematrix if and only if P = 0 (by Theorem 4.2 (d)), some characterizations of the CCE-inverse matrixare derived as follows. Theorem 4.3.
Let H ∈ C n × nk have the form of (2 . . Then the following assertions are equivalent:(a) H is a CCE-inverse matrix;(b) H C, †(cid:13) = H D ;(c) H C, †(cid:13) = H † ,D ; d) H C, †(cid:13) = H w (cid:13) ;(e) H k +1 H C, †(cid:13) = H k ;(f ) H k +1 H †(cid:13) = H k ;(g) HH C, †(cid:13) = HH D ;(h) H k H C, †(cid:13) = H k H †(cid:13) ;(i) [ HH C, †(cid:13) , H C, †(cid:13) H ] = 0 .Proof. According to Theorem 4.2, we get that H is a CCE-inverse matrix if and only if P = 0. If P = 0, then H C, †(cid:13) = H D = H † ,D = H †(cid:13) = H w (cid:13) = U T −
00 0 U ∗ . Thus it can be easily seenthat ( a ) ⇒ ( b ), ( a ) ⇒ ( c ), ( a ) ⇒ ( d ), ( a ) ⇒ ( e ), ( a ) ⇒ ( f ), ( a ) ⇒ ( g ), ( a ) ⇒ ( h ) and ( a ) ⇒ ( i ).“( b ) ⇒ ( a )”. If H C, †(cid:13) = H D , by (3 .
8) and (3 . T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 = T − ( T k +1 ) − e T , then, ( T k +1 ) − e T = 0 ⇒ e T = 0 ⇒ P = 0.“( c ) ⇒ ( a )”. If H C, †(cid:13) = H † ,D , it follows from (3 .
12) and (3 .
13) that e T = 0 ⇒ P = 0.“( d ) ⇒ ( a )”. If H C, †(cid:13) = H w (cid:13) , due to (3 .
12) and Theorem 3.1 of [27], we obtain T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 = T − T − P , which implies P = 0.“( e ) ⇒ ( a )”. According to H k +1 H C, †(cid:13) = H k , we deduce that T k +1 T e T T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 = T k e T . Thus, e T = 0 which lead to P = 0.“( f ) ⇒ ( a )”. The proof is similar to that of ( e ) ⇒ ( a ).“( g ) ⇒ ( a )”. Assume that HH C, †(cid:13) = HH D . Then P H k = HH D and P H k = HH D ⇒ U I t
00 0 U ∗ = U T P T − ( T k +1 ) − e T U ∗ . T − k e T = 0 ⇒ e T = 0 ⇒ P = 0.“( h ) ⇒ ( a )”. The proof is similar to that of ( g ) ⇒ ( a ).“( i ) ⇒ ( a )”. If [ HH C, †(cid:13) , H C, †(cid:13) H ] = 0, it follows from the fact that HH C, †(cid:13) = P H k and[ HH C, †(cid:13) , H C, †(cid:13) H ] = 0 that U I t
00 0 T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0
T P Q U ∗ = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0
T P Q I t
00 0 U ∗ . Thus, T ∗ Λ T T ∗ Λ P = T ∗ Λ T I n − t − Q † Q ) P ∗ Λ T , which implies that P = 0 . In [9], the authors introduced the following classes of matrices C k − EP n , C k, †(cid:13) n , C k,D † n , C k, † Dn , C k,C † n , C k, w (cid:13) n . Similarly, we establish the concept of k − CCE-inverse matrix and propose the characterizationsof k − CCE inverse matrix as follows.
Definition 4.4.
Let H ∈ C n × nk . We say that H is a k-CCE inverse matrix if H ∈ C k,C †(cid:13) n and definethe set of k -CCE inverse matrices by C k,C †(cid:13) n = { H | H ∈ C n × nk , H k H C, †(cid:13) = H C, †(cid:13) H k } . Now, we give the characterizations of k − CCE-inverse matrices and prove that the k − CCE inversematrix is the same as the CCE-inverse matrix.
Theorem 4.5.
Let H ∈ C n × nk have the form of (2 . . Then the following assertions are equivalent:(a) H ∈ C k,C †(cid:13) ;(b) H is a CCE-inverse matrix;(c) H t H C, †(cid:13) = H C, †(cid:13) H t ( t ≥ k ) ;(d) H m H C, †(cid:13) = H C, †(cid:13) H m ( m ∈ N ) . roof. “( a ) ⇔ ( b )”. By (2 . H k = U T k e T U ∗ , where e T = T k − P + T k − P Q + · · · + T P Q k − + P Q k − . Observe that H C, †(cid:13) = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 U ∗ ,H k H C, †(cid:13) = H k H † P H k = H k − P H k = U T k −
00 0 U ∗ ,H C, †(cid:13) H k = H † P H k H k = H † H k = U T ∗ Λ T k T ∗ Λ e T ( I n − t − Q † Q ) P ∗ Λ T k ( I n − t − Q † Q ) P ∗ Λ e T U ∗ . Hence, H k H C, †(cid:13) = H C, †(cid:13) H k if and only if e T = 0 and ( I n − t − Q † Q ) P ∗ = 0, that is to say that P = 0.“( b ) ⇔ ( c )”. The proof is analogous to that of ( a ) ⇔ ( b ).“( b ) ⇔ ( d )”. Noting that H = HH †(cid:13) H = U T P U ∗ , we derive that H m = U T m T m − P U ∗ , H m H C, †(cid:13) = U T m T ∗ Λ + T m − P ( I n − t − Q † Q ) P ∗ Λ 00 0 U ∗ ,H C, †(cid:13) H m = U T ∗ Λ 0( I n − t − Q † Q ) P ∗ Λ 0 T m T m − P U ∗ = U T ∗ Λ T m T ∗ Λ T m − P ( I n − t − Q † Q ) P ∗ Λ ( I n − t − Q † Q ) P ∗ Λ T m − P U ∗ , where Λ = ( T T ∗ + P ( I n − t − Q † Q ) P ∗ ) − , t = r ( H k ) . Therefore, H C, †(cid:13) H m = H m H C, †(cid:13) if and onlyif P = 0. The authors would like to appreciate Pro. Dragana of Cihu scholars in Hubei Normal universityfor her precious comments. 21 eferences [1] O.M. Baksalary, G.P. Styan and G. Treakler. On a matrix decomposition of Hartwig andSpindelb¨ock.
Linear Algebra and its Applications , 430: 2798-2812, 2009.[2] O.M. Baksalary and G. Trenkler. On a generalized core inverse.
Applied Mathematics andComputation , 236: 450-457, 2014.[3] O.M. Baksalary and G. Trenkler. Core inverse of matrices.
Linear and Multilinear Algebra , 58:681-697, 2010.[4] J. Benitez, E. Boasso, H.W. Jin. On one-sided (B,C)-inverse of arbitrary matrices.
ElectronicJournal of Linear Algebra , 32: 391-422, 2017.[5] A. Ben-Israel and T.N.E. Greville.
Generalized Inverses: Theory and applications, second ed.
Springer-Verlag, New-York, 2003.[6] J.F. Cai, S. Osher and Z.W Shen. Linearized bregman iterations for compressed sensing.
Math-ematics of Computation , 78 (267), 1515-1536, 2009.[7] M.P. Drazin. A class of outer generalized inverses.
Linear Algebra and its Applicatians , 436:1909-1923, 2012.[8] M.P. Drazin. Pseudo-Inverses in Associative Rings and Semigroups.
American MathematicalMonthly , 65(7): 506-514, 1958.[9] C.Y. Deng, H.K. Du. Representation of the Moore-Penrose inverse of 2 × Journal of the Korean Mathematical Society , 46: 1139-1150, 2009.[10] D.E. Ferreyra, F.E. Levis, N. Thome. Maximal classes of matrices determining generalizedinverse.
Applied Mathematics and Computatian , 333: 42-52, 2018.[11] D.E. Ferreyra, F.E. Levis, N. Thome. Revisiting the core-EP inverse and its extension torectangular matrices.
Quaestiones Mathematicae , 41: 1-17, 2018.[12] D.E. Ferreyra, F.E. Levis, N. Thome. Characterizations of k-commutative egualities for someouter generalized inverse.
Linear and Multilinear Algebra , 68(1): 177-192, 2020.2213] R.E. Hartwig, K. Spindelb¨ock. Matrices for which A ∗ and A † commute. Linear and MultilinearAlgebra , 14: 241-256, 1984.[14] H. Kurata. Some theorems on the core inverse of matrices and the core partial ordering.
AppliedMathematics and Computatian , 316: 43-51, 2018.[15] G.J. Luo, K.Z. Zuo and L. Zhou. Revisitation of core inverse.
Wuhan University Journal ofNatural Sciences , 20: 381-385, 2015.[16] J.L. Mcclelland and D.E. Rumelhart.
Explorations in parallel distributed processing: a handbookof models, programs, and exercises.
Cambridge Mit Press, 1986.[17] S.K. Mitra, P. Bhimasankaram and S.B. Malik. Matrix Partial Orders, Shorted Operators andApplications.
World Scientific , 2010.[18] S.B. Malik, L. Rueda and N. Thome. The class of m-EP and m-normal matrices.
Linear andMultilinear Algebra , 64(11): 2119-2132, 2016.[19] M. Mehdipour, A. Salemi. On a new generalized inverse of matrices.
Linear and MultilinearAlgebra , 66: 1-8, 2018.[20] S.B. Malik, N. Thome. On a new generalized inverse for matrices of an arbitrary index.
AppliedMathematics and Computation , 226: 575-580, 2014.[21] K.M. Prasad, K.S. Mohana. Core-EP inverse.
Linear and Multilinear Algebra , 62: 792-802,2014.[22] R.A. Penrose. A Generalized Inverse for Matrices.
Mathematical Proceedings of the CambridgePhilosophical Society , 51(03): 406-413, 1955.[23] C. R. Rao and S. K. Mitra.
Generalized Inverse of Matrices and its Applications.
Wiley, NewYork, 1971.[24] D.S. Rak´c, N.C. Dinˇci´c, D.S. Djordjevi´c. Core inverse and core partial order of Hilbert spaceoperators.
Applied Mathematics and Computation , 244: 283-302, 2014.[25] P.S. Stanimirovi´c, D.S. Cvetkovi´c-Ili´c, S. Miljkovi´c, M. Miladinovi´c. Full-rank representationsof { , } , { , } -inverses and successive matrix squaring algorithm. Applied Mathematics andComputation , 217(22), 9358-9367, 2011. 2326] H.X. Wang. Core-EP decomposition and its applications.
Linear Algebra and its Applicatians ,508: 289-300, 2016.[27] H.X. Wang, J.L. Chen. Weak group inverse.
Open Mathematics , 16(1): 1218-1232, 2018.[28] H.X. Wang, X.J. Liu. Partial orders based on core-nilpotent decomposition.
Linear Algebra andits Applications , 488: 235-248, 2016.[29] H.X. Wang, X.J. Liu. The weak group matrix.
Aequationes Mathematicae ,93(6): 1261-1273,2019.[30] X.N. Wang, C.Y. Deng. Properties of m − EP operators.
Linear and Multilinear Algebra , 65(7):1349-1361, 2017.[31] S.Z. Xu, J.L. Chen, D. Mosi´c. New characterizations of the CMP inverse of matrices.
Linearand Multilinear Algebra , 68(4): 790-804, 2020.[32] Y.X. Yuan, K.Z. Zuo. Compute lim λ → X ( λI p + Y AX ) − Y be the product singular value decom-position. Linear and Multilinear Algebra , 64: 269-278, 2016.[33] K.Z. Zuo,Y.J. Cheng. The new revisitation of core-EP inverse of matrices.
Filomat , 33(10):3061-3072, 2019.[34] K.Z. Zuo, D.S. Cvetkovi´c-Ili´c, Y.J. Cheng. Different characterizations of DMP-inverse of matri-ces.