aa r X i v : . [ m a t h . R A ] F e b Relative EP matrices
D.E. Ferreyra ∗ and Saroj B. Malik † Abstract
The purpose of the present work is to introduce the concept of relative EP matrix ofa rectangular matrix relative to a partial isometry (or, in short, T -EP matrix) hithertounknown. We extend various basic results on EP matrices and we study the relationshipbetween T -hermitian, T -normal and T -EP matrices. The main theorems of this paperconsist in providing canonical forms of relative EP matrices when matrices involved arerectangular as well as square. We then use them to characterize the relative EP matricesand show their properties. In fact, an interesting fact that has emerged is that A is T -EPif and only if there is an EP matrix C such that A = CT and C = T T ∗ C whatever be thematrix, square or rectangular. We also give various necessary and sufficient conditions fora matrix to be T -EP. AMS Classification: 15A09; 15A27; 15B57.Keywords: Moore-Penrose inverse, Partial isometry, EP matrix, T -EP matrix. One of the problems in Matrix Theory is to study and analyze the many classes of special typematrices for their properties (algebraic as well as geometric). The class of EP (range-hermitian)matrices is one such class. This class has attracted the attention of several authors since theywere first defined by H. Schwerdtfeger [16]. Amongst many references available some of themare [1, 2, 3, 4, 12, 14]. One of the reasons for interest in this class is the fairly weaker conditionon the matrix, namely the equality of it’s range space and the range space of its conjugatetranspose. EP matrices include in them the wide classes of matrices as special cases suchas hermitian matrices, skew-hermitian matrices, unitary matrices, normal matrices, and alsononsingular matrices.The EP matrices have been generalized in many ways. The first such generalization knownas bi-EP matrices is given by Hartwig and Spindelb¨ock in [7]. Since then on several of theirgeneralizations have appeared some of them being conjugate EP matrix [10], k -EP [11], weightedEP matrices [17], m -EP [9], k -core EP [5], and k -DMP [5]. All these extensions have beendefined on the set of complex square matrices. Our aim here in this work is to extend the classof EP matrices to a class of relative EP matrices when matrices are rectangular matrices. ∗ Universidad Nacional de R´ıo Cuarto, CONICET, FCEFQyN, RN 36 KM 601, 5800 R´ıo Cuarto, C´ordoba,Argentina. E-mail: [email protected] † School of Liberal Studies, Ambedkar University, Kashmere Gate, Delhi, India. E-mail: [email protected]
1e introduce in this paper the concept of relative EP matrix for a rectangular matrix A, extending the notion of square EP matrices to rectangular matrices. We recall that theconcepts of relative hermitian and relative normal matrices were introduced by Hestenes in [8]in order to develop a spectral theory for rectangular matrices. Motivated by these definitions,we introduce in this paper the concept of relative EP matrix for a rectangular matrix A, (inshort T -EP matrices, where T is a fixed partial isometry). Like in classical case, the relativeEP matrices are a generalization of each of relative hermitian and relative normal matrices.It is shown that every relative normal matrix is relative EP but not conversely. Several basicproperties of EP matrices extend to relative EP matrices.The main theorems of this paper are the development of canonical forms of relative EPmatrices and using these canonical forms we give various characterizations of these matrices.The paper is organized as follows. In Section 2, we introduce and characterize the concept ofrelative EP matrix as a generalization of the concept of EP matrix to the rectangular matrices.This new class of matrices is called the class of T -EP matrices and contains the class of relativenormal matrices and therefore the class of relative hermitian matrices defined in [8]. We alsoobtain here some properties of these matrices. Section 3 consists of obtaining canonical formsof a relative EP matrix where the matrix can be rectangular or square. The advantage of thecanonical form in either case is that they reveal the relationship of a T -EP matrix A to anEP matrix necessarily other than A (when T is not the trivial partial isometry I n ). Usingthese canonical forms we derive some characterizations of T -EP matrices. In particular, werecover various well-known results for EP matrices obtained by Pearl in [14]. Section 4 givesthe interrelations between EP and T -EP matrices as well as some more characterizations of T -EP matrices when T belongs to certain matrix class. The final Section 5 deals with theproblem of the sum of two relative EP matrices.We denote the set of all m × n complex matrices by C m × n . For A ∈ C m × n , the symbols A ∗ ,rank( A ), N ( A ), and R ( A ) will stand for the conjugate transpose, the rank, the null space, andthe range space of A , respectively. For A ∈ C n × n , A − will denote the inverse of A and I n willrefer to the n × n identity matrix.The symbol A † denotes the Moore-Penrose inverse of a matrix A ∈ C m × n , which is theunique matrix satisfying the following four Penrose conditions [2]: AA † A = A, A † AA † = A † , ( AA † ) ∗ = AA † , ( A † A ) ∗ = A † A. A matrix A ∈ C n × n is an EP matrix if R ( A ) = R ( A ∗ ), or equivalently AA † = A † A , see,e.g. [2, 14]. Recall that T ∈ C m × n is a partial isometry if it verifies T = T T ∗ T . Note that T isa partial isometry if and only if T † = T ∗ [2, Theorem 4.2.1]. Throughout this work we will use T ∈ C m × n as a fixed partial isometry (We note that T can also be a square matrix, but then itwill be clear from the context of the usage). T -EP matrices In this section we introduce the concept of relative EP matrix for a matrix A ∈ C m × n withrespect to a partial isometry T ∈ C m × n . We present various equivalent characterizations of thisnew class of matrices and we show that this class of matrices contains the following two classesof matrices defined by Hestenes [8]: 2 efinition 2.1. Let
A, T ∈ C m × n . Then A is called relative hermitian to T (or, in short, T -hermitian) if A = T A ∗ T. Definition 2.2.
Let
A, T ∈ C m × n . We say that A is relative normal to T (or, in short, T -normal) if A = T T ∗ A = AT ∗ T and AA ∗ T = T A ∗ A . Like in classical case as is known that every hermitian matrix is normal, we show that every T -normal matrix is T -hermitian. Theorem 2.3.
Let
A, T ∈ C m × n . If A is T -hermitian then A is T -normal.Proof. Since A is T -hermitian, A = T A ∗ T. So, AT ∗ T = ( T A ∗ T ) T ∗ T = T A ∗ ( T T ∗ T ) = T A ∗ T = A. Similarly,
T T ∗ A = T T ∗ ( T A ∗ A ) = A . Also, AA ∗ T = ( T A ∗ T ) A ∗ T = T A ∗ ( T A ∗ T ) = T A ∗ A. Thus both (i) and (ii) in Definition 2.2 hold.We now define the concept of relative EP matrix respect to a partial isometry T as follows: Definition 2.4.
Let
A, T ∈ C m × n . Then we say A is relative EP to T (or, in short, T -EP) if R ( A ) = R ( T A ∗ T ) and A = AT ∗ T . Remark 2.5.
In particular, when the partial isometry T is the identity matrix above definitionreduces to R ( A ) = R ( A ∗ ), which implies that A is an EP matrix [14].We next show that a T -normal matrix is a T -EP matrix. Theorem 2.6.
Let
A, T ∈ C m × n If A is T -normal then A is T -EP.Proof. Let A be T -normal. Then A = AT ∗ T = T T ∗ A and AA ∗ T = T A ∗ A (or equivalently T ∗ AA ∗ = A ∗ AT ∗ ). Clearly, A = AT ∗ T . It remains to see that R ( A ) = R ( T A ∗ T ). In fact, R ( A ) = R ( AA ∗ )= R ( T T ∗ AA ∗ )= R ( T A ∗ AT ∗ )= R ( T A ∗ T T ∗ AT ∗ )= R ( T A ∗ T ( T A ∗ T ) ∗ )= R ( T A ∗ T ) . We now give a few examples. Just as an observation, if the matrices involved are squarematrices, every EP matrix A ∈ C n × n is relative EP, relative to trivial partial isometry I n . Example 2.7.
A trivial example of a T -EP matrix is the partial isometry T itself because T = T T ∗ T . xample 2.8. Let T = , then T is a partial isometry and all T -hermitian matricesare of the type A = a z z b , where a, b are real numbers and z ∈ C . Note that A is nothermitian. Moreover, A is not EP. Example 2.9.
Let T be the partial isometry in Example 2.8. Let A = . By usingprevious example, it is clear that A is T -hermitian. Therefore, by Theorems 2.3 and 2.6 wehave that A is T -normal and T -EP. However, note that A is not EP. Our next example shows that the converse of Theorem 2.6 is false.
Example 2.10.
Let T = . Then T is a partial isometry. Let A = . Then A is T -EP, but not T -normal. The following two lemmas are fundamental to developing the properties and characteriza-tions of relative EP matrices. We also need to use the following important relation between thenull space of a matrix A ∈ C m × n and the range space of its conjugate transpose A ∗ , namely R ( A ) = N ( A ∗ ) ⊥ , (2.1)where the superscript ⊥ denotes the orthogonal complement of N ( A ∗ ). Lemma 2.11.
Let
A, T ∈ C m × n . Then the following are equivalent: (i) A = T T ∗ A ; (ii) R ( A ) ⊆ R ( T ) ; (iii) N ( T ∗ ) ⊆ N ( A ∗ ) ; (iv) A † = A † T T ∗ .Proof. (i) ⇒ (ii). Easy.(ii) ⇒ (iii). Using (2.1) in R ( A ) ⊆ R ( T ) implies N ( A ∗ ) ⊥ ⊆ N ( T ∗ ) ⊥ , which yields to N ( T ∗ ) ⊆N ( A ∗ ).(iii) ⇒ (iv). As N ( A ∗ ) = N ( A † ) we have N ( T ∗ ) ⊆ N ( A † ) and is further equivalent to A † = A † ( T ∗ ) † T ∗ . As T † = T ∗ , we have A † = A † ( T ∗ ) ∗ T ∗ = A † T T ∗ .(iv) ⇒ (i). Pre-multiplying A † = A † T T ∗ by A ∗ A, we obtain A ∗ = A ∗ T T ∗ and therefore, A = T T ∗ A by taking ∗ on both sides.Similarly one can show the following: Lemma 2.12.
Let
A, T ∈ C m × n . Then the following are equivalent: A = AT ∗ T ; (ii) N ( T ) ⊆ N ( A ) ; (iii) R ( A ∗ ) ⊆ R ( T ∗ ) ; (iv) A † = T ∗ T A † . We now use these lemmas to obtain some characterizations and properties of T -EP matrices. Theorem 2.13.
Let
A, T ∈ C m × n . Then the following are equivalent: (i) A is T -EP (i.e., R ( A ) = R ( T A ∗ T ) and A = AT ∗ T ); (ii) R ( A ) = R ( T A ∗ T ) and N ( T ) ⊆ N ( A ) ; (iii) N ( A ∗ ) = N ( T ∗ AT ∗ ) and R ( A ∗ ) ⊆ R ( T ∗ ) ; (iv) R ( A ) = R ( T A ∗ ) and N ( T ) ⊆ N ( A ) ; (v) R ( A ) = R ( T A ∗ ) and A = AT ∗ T ; (vi) N ( A ∗ ) = N ( AT ∗ ) and R ( A ∗ ) ⊆ R ( T ∗ ) .Proof. (i) ⇔ (ii). It follows from Definition 2.4 and Lemma 2.11.(ii) ⇔ (iii). It is a direct consequence of (2.1).(ii) ⇒ (iv). We only need to prove that R ( A ) = R ( T A ∗ ). Since R ( A ) = R ( T A ∗ T ) weobtain R ( T A ∗ T ) ⊆ R ( T A ∗ ) and rank( A ) = rank( T A ∗ T ). Moreover, rank( T A ∗ ) ≤ rank( A ∗ ) =rank( A ) = rank( T A ∗ T ). Therefore, R ( A ) = R ( T A ∗ ).(iv) ⇒ (ii). Suppose R ( A ) = R ( T A ∗ ) and N ( T ) ⊆ N ( A ). Only we need to show that R ( A ) = R ( T A ∗ T ). In fact, clearly R ( T A ∗ T ) ⊆ R ( A ). Also, as R ( A ) ⊆ R ( T ) implies T T ∗ A = A by Lemma 2.11 (i). So, rank( A ) = rank( T A ∗ ) = rank( T A ∗ T T ∗ ) ≤ rank( T A ∗ T ),whence R ( A ) = R ( T A ∗ T ).(iv) ⇔ (v) By Lemma 2.12.(v) ⇔ (vi) Follows from (2.1) and Lemma 2.12. Theorem 2.14.
Let A ∈ C m × n be a T -EP matrix, then the following hold: (i) A = T T ∗ A . (ii) rank( A ) = rank( T A ∗ T ) = rank( T A ∗ ) = rank( A ∗ T ) ; (iii) ( T ∗ A ) † = A † T ; (iv) ( AT ∗ ) † = T A † ; (v) ( T ∗ AT ∗ ) † = T A † T .Proof. (i) As A is T -EP, we have R ( A ) = R ( T A ∗ T ), and so R ( A ) ⊆ R ( T ). Now, Lemma 2.11implies A = T T ∗ A .(ii) By definition of T -EP matrix it is clear that rank( A ) = rank( T A ∗ T ). Thus,rank( A ) = rank( T A ∗ T ) ≤ rank( T A ∗ ) ≤ rank( A ) = rank( T A ∗ T ) ≤ rank( A ∗ T ) ≤ rank( A ) , A ) = rank( T A ∗ T ) = rank( T A ∗ ) = rank( A ∗ T ).(iii) We check this by direct verification. Let X := A † T . By definition of T -EP matrix andpart (i) we have A = AT ∗ T and A = T T ∗ A , respectively. Thus, T ∗ AXT ∗ A = T ∗ AA † T T ∗ A = T ∗ AA † A = T ∗ A,XT ∗ AX = A † T T ∗ AA † T = A † AA † T ∗ = A † T ∗ = X, ( T ∗ AX ) ∗ = ( T ∗ AA † T ) ∗ = T ∗ ( AA † ) ∗ T = T ∗ AA † T = T ∗ AX, ( XT ∗ A ) ∗ = ( A † T T ∗ A ) ∗ = ( A † A ) ∗ = A † A = A † T T ∗ A = XT ∗ A. By uniqueness of the Moore-Penrose inverse we have X = ( T ∗ A ) † .Items (iv) and (v) can be proved along lines of (iii). T -EP matrices and consequences In this section we exhibit the main results of this paper namely, canonical forms of a T -EPmatrix A . We first derive a canonical form of a T -EP matrix, when the matrix is rectangular.The tool we use is the Singular Value Decomposition (SVD). As a special case we have acanonical form of a square matrix. However, we provide another canonical form of a squarematrix by using the Hartwig-Spindelb¨ock decomposition of a square matrix, which by itself wasderived by using SVD. The reason to include this is that it has its own merits. We then discussimmediate consequences of both the canonical forms. We also give some characterizations of T -EP matrices. Theorem 3.1. (SVD) Let A ∈ C m × n a nonnull matrix of rank r > and let σ ≥ σ ≥ · · · ≥ σ r > be the singular values of A . Then there exist unitary matrices U ∈ C m × m and V ∈ C n × n such that A = U (cid:20) Σ 00 0 (cid:21) V ∗ , where Σ = diag ( σ , σ , . . . , σ r ) . In particular, the Moore-Penroseinverse of A is given by A † = V (cid:20) Σ −
00 0 (cid:21) U ∗ . (3.1) Theorem 3.2.
Let
A, T ∈ C m × n be such that rank( A ) = r . Then the following are equivalent. (i) A is T -EP; (ii) There exists nonsingular matrix D of order r and a unitary matrix U such that A = U (cid:20) D
00 0 (cid:21) U ∗ T and T = U (cid:20) T T (cid:21) V ∗ , (3.2) where T and T are matrices satisfying T T ∗ = I r and T T ∗ T = T . (iii) There exists an EP matrix E ∈ C m × m such that A = ET and T T ∗ E = E .Proof. (i) ⇒ (ii). Let A be T -EP. By Theorem 3.1, there exist unitary matrices U ∈ C m × m and V ∈ C n × n , and a nonsingular matrix Σ such that A = U (cid:20) Σ 00 0 (cid:21) V ∗ . Since A is T -EP, Theorem6.13 (v) implies R ( A ) = R ( T A ∗ ) and A = AT ∗ T . Let T be partitioned as T = U (cid:20) T T T T (cid:21) V ∗ ,such that the product T A ∗ and AT ∗ are defined. Clearly, A = AT ∗ T is equivalent to (cid:20) Σ 00 0 (cid:21) = (cid:20) Σ 00 0 (cid:21) (cid:20) T ∗ T + T ∗ T T ∗ T + T ∗ T T ∗ T + T ∗ T T ∗ T + T ∗ T (cid:21) , equivalently, T ∗ T + T ∗ T = I r , (3.3) T ∗ T + T ∗ T = 0 . (3.4)Since T is a partial isometry, i.e., T = T T ∗ T , using (3.3) and (3.4) we obtain (cid:20) T T T T (cid:21) = (cid:20) T T T T (cid:21) (cid:20) I r T ∗ T + T ∗ T (cid:21) , whence T = T T ∗ T + T T ∗ T , (3.5) T = T T ∗ T + T T ∗ T . (3.6)Also, as R ( A ) = R ( T A ∗ ) we get AA † T A ∗ = T A ∗ . By using (3.1), after simple matrix compu-tations we obtain (cid:20) T Σ ∗
00 0 (cid:21) = (cid:20) T Σ ∗ T Σ ∗ (cid:21) , whence T = 0 because Σ is nonsingular. In consequence, from (3.3) we obtain T ∗ T = I r orequivalently T T ∗ = I r as T is a square matrix of order r . Therefore, (3.4) implies T = 0. So,from (3.6) we get T = T T ∗ T i.e., T is a partial isometry.Finally, once again from A = AT ∗ T we have A = U (cid:20) Σ T ∗
00 0 (cid:21) U ∗ T = U (cid:20) D
00 0 (cid:21) U ∗ T, where D is the matrix Σ T ∗ which clearly is nonsingular.(ii) ⇒ (iii). Write E = U (cid:20) D
00 0 (cid:21) U ∗ , where D is nonsingular. Clearly E is EP and A = ET .Also, since T T ∗ = I r , it follows that T T ∗ E = U (cid:20) I r T T ∗ (cid:21) (cid:20) D
00 0 (cid:21) U ∗ = U (cid:20) D
00 0 (cid:21) U ∗ = E. (iii) ⇒ (i). We assume A = ET and T T ∗ E = E . As E is EP then R ( E ) = R ( E ∗ ) . Thus, R ( T A ∗ ) = R ( T T ∗ E ∗ ) = T T ∗ R ( E ∗ ) = T T ∗ R ( E ) = R ( T T ∗ E ) = R ( E ) , whence R ( A ) = R ( ET ) ⊆ R ( E ) = R ( T A ∗ ).Moreover, rank( AT ∗ ) ≤ rank( A ) . Thus, R ( A ) = R ( T A ∗ ). Also, as A = ET we get N ( T ) ⊆N ( ET ) = N ( A ). Thus, by Theorem 2.14 (iv) A is T -EP. Remark 3.3.
Note that under any one of equivalent conditions (i)-(iii) in Theorem 3.2 wehave
ET T ∗ = E . 7heorem 3.2 (iii) enables us to obtain the following formula for the Moore-Penrose inverseof a matrix. Theorem 3.4.
Let A , T , and E be as in Theorem 3.2. Then the Moore-Penrose inverse of A is given by A † = T ∗ E † . (3.7) Proof.
We check this by direct verification. Let X := T ∗ E † . As is T -EP, Theorem 3.2 andRemark 3.3 imply A = ET and E = T T ∗ E = ET T ∗ . Therefore, AXA = ET ( T ∗ E † ) ET = ( ET T ∗ ) E † ET = ET = A,XAX = T ∗ E † ( ET T ∗ ) E † = T ∗ E † EE † = T ∗ E † = X. Also we have AX = ( ET T ∗ ) E † = EE † and XA = T ∗ E † ET . Clearly, AX and XA arehermitian matrices. So, by uniqueness of the Moore-Penrose inverse we have X = A † .Some important consequences of Theorem 3.2 are the following: Corollary 3.5.
Let A , T , D , and U be as in Theorem 3.2. Then the Moore-Penrose inverseof A is given by A † = T ∗ U (cid:20) D −
00 0 (cid:21) U ∗ . (3.8) Proof.
It directly follows from (3.7) by using the fact that E = U (cid:20) D
00 0 (cid:21) U ∗ . Corollary 3.6.
Let
A, T ∈ C m × n be such that rank( A ) = r . If A is T -EP, then there exists apartial isometry R ∈ C n × n and a nonsingular matrix D of order r such that T ∗ A = R (cid:20) D
00 0 (cid:21) R ∗ . (3.9) Proof.
Let A be T -EP. By Theorem 3.2, there exists a nonsingular matrix D of order r and aunitary matrix U such that A = U (cid:20) D
00 0 (cid:21) U ∗ T . Pre-multiplying the previous equality by T ∗ and by taking R := T ∗ U we obtain (3.9). Clearly, R is a partial isometry. Theorem 3.7.
Let
A, T ∈ C m × n . Then A is T -EP if and only if T A † A = AA † T and A = AT ∗ T .Proof. In order to prove the necessity of the condition it is enough to show that
T A † A = AA † T because clearly A = AT ∗ T holds by definition of T -EP matrix. Thus, by Theorem 3.2 andRemark 3.3, there exists an EP matrix E such that A = ET and E = T T ∗ E = ET T ∗ .Moreover, (3.7) implies A † = T ∗ E † . Consequently, as EE † = E † E we obtain AA † T = ET T ∗ E † T = EE † T = T T ∗ EE † T = T T ∗ E † ET = T A † A. Conversely, we assume that
T A † A = AA † T and A = AT ∗ T hold. Consider T A † A = AA † T. Multiplying on right by A ∗ , we have T A ∗ = AA † T A ∗ . This implies R ( T A ∗ ) ⊆ R ( A ). Sincerank( A ) = rank( AT ∗ T ) ≤ rank( AT ∗ ) = rank(( AT ∗ ) ∗ ) = rank( T A ∗ ), we get R ( A ) = R ( T A ∗ ) . Now, from Theorem 2.13 (v) it follows that A is T -EP.8 orollary 3.8. Let
A, T ∈ C m × n . Then A is T -EP if and only if T A † A = AA † T and any oneof equivalent conditions (i)-(iv) in Lemma 2.12 hold. Theorem 3.9.
Let
A, T ∈ C m × n . Then the following are equivalent: (i) A is T -EP; (ii) A ∗ is T ∗ -EP; (iii) A † is T ∗ -EP.Proof. (i) ⇒ (ii). By Theorem 3.7, it is enough to establish T ∗ ( A ∗ ) † A ∗ = A ∗ ( A ∗ ) † T ∗ and A ∗ = A ∗ ( T ∗ ) ∗ T ∗ = A ∗ T T ∗ . As A is T -EP, from Theorem 3.7 we have AA † T = T A † A and A = AT ∗ T .Moreover, Theorem 2.14 (i) implies A = T T ∗ A , and by taking ∗ on both sides we have A ∗ = A ∗ T T ∗ . It remains to prove that T ∗ ( A ∗ ) † A ∗ = A ∗ ( A ∗ ) † T ∗ but this equality is true if and onlyif AA † T = T A † A . This shows (ii).(ii) ⇒ (i). It is similar to the previous proof by interchanging the roles of A and A ∗ .(i) ⇒ (iii). By Theorem 3.7, we know that A is T -EP implies AA † T = T A † A and A = AT ∗ T .Moreover, Theorem 2.14 (i) implies A = T T ∗ A , which is equivalent to A † = A † T T ∗ by Lemma2.11. As T A † A = AA † T, taking ∗ on both sides, we have A † AT ∗ = T ∗ AA † . Using A = ( A † ) † we have A † ( A † ) † T ∗ = T ∗ ( A † ) † A † . So, once again Theorem 3.7 implies that A † is T ∗ -EP.(iii) ⇒ (i). Along lines of (ii) ⇒ (i). Theorem 3.10.
Let
A, T ∈ C m × n . Then the following are equivalent: (i) A is T -EP; (ii) AT ∗ is EP and A = AT ∗ T ; (iii) T A ∗ is EP and A = AT ∗ T ; (iv) T A † is EP and A = AT ∗ T .Proof. (i) ⇒ (ii). We assume A is T -EP. Then by Theorem 3.2 and Remark 3.3, there exists anEP matrix E such that A = ET and E = ET T ∗ = T T ∗ E . Therefore AT ∗ = ET T ∗ = E , thatis, AT ∗ is EP. Clearly, A = AT ∗ T by definition of T -EP matrix.(ii) ⇒ (i). We consider AT ∗ is EP and A = AT ∗ T . Taking E = AT ∗ , from A = AT ∗ T we have A = ET , where E is clearly an EP matrix. Moreover, as R ( E ) = R ( E ∗ ) we have R ( E ) ⊆ R ( T ), which is equivalent to T T ∗ E = E . Now, by using Theorem 3.2 it follows that A is T -EP.(ii) ⇔ (iii). It is a direct consequence of the fact that a square complex matrix B is EP if andonly if B ∗ is EP.(i) ⇒ (iv). We assume A is T -EP. By Theorem 3.11 (v) we have ( AT ∗ ) † = T A † . Since a matrix B is EP if and only B † is EP, we have the equivalence.Some more properties of T -EP matrices are established in the following theorem: Theorem 3.11.
Let A ∈ C m × n be a T -EP matrix, then the following hold: (i) ( T A ∗ ) † = ( A ∗ ) † T ∗ ; (ii) ( A ∗ T ) † = T ∗ ( A ∗ ) † ; T A ∗ T ) † = T ∗ ( A ∗ ) † T ∗ ; (iv) ( T A † ) † = AT ∗ ; (v) ( A † T ) † = T ∗ A ; (vi) ( T A † T ) † = T ∗ AT ∗ .Proof. It is a direct consequence from Theorem 2.14 and Theorem 3.9.We note that for square matrices we can obtain a canonical form from Theorem 3.2 bysimply taking m = n. However, we present another canonical form a square matrix A for whichwe use the famous Hartwig-Spindelb¨ock decomposition [7](which it self was derived from SVDof A ). For this , we assume all matrices are square from now onwards and for the next sectionof this paper. To give this canonical form we need the following two results. Theorem 3.12. [7, Hartwig-Spindelb¨ock decomposition] Let A ∈ C n × n of rank r > . Thenthere exists a unitary matrix U ∈ C n × n such that A = U (cid:20) Σ K Σ L (cid:21) U ∗ , (3.10) where Σ = diag ( σ I r , σ I r , . . . , σ t I r t ) is the diagonal matrix of singular values of A , σ > σ > · · · > σ t > , r + r + · · · + r t = r , and K ∈ C r × r , L ∈ C r × ( n − r ) satisfy KK ∗ + LL ∗ = I r . Theorem 3.13. [1] Let A ∈ C n × n be a matrix written as in (3.10). Then A † = U (cid:20) K ∗ Σ − L ∗ Σ − (cid:21) U ∗ , AA † = U (cid:20) I r
00 0 (cid:21) U ∗ . Theorem 3.14.
Let
A, T ∈ C n × n , T a partial isometry, and r = rank( A ) . Then the followingare equivalent. (i) A is T -EP; (ii) There exists nonsingular matrix D of order r and a unitary matrix U such that A = U (cid:20) D
00 0 (cid:21) U ∗ T ; where T = U (cid:20) T T T T (cid:21) U ∗ with T T ∗ + T T ∗ = I r and T T ∗ + T T ∗ = 0 . (iii) There exists an EP matrix C such that A = CT and T T ∗ C = C .In particular, under any one of equivalent conditions (i)-(iii) we have CT T ∗ = C .Proof. We show (i) ⇒ (ii) ⇒ (iii) ⇒ (i).(i) ⇒ (ii). Let A be T -EP. By Theorem 2.13 (v) R ( A ) = R ( T A ∗ ) and A = AT ∗ T. We write A asin (3.10). Let T be partitioned in conformation with the partition of A as T = U (cid:20) T T T T (cid:21) U ∗ . Then A = AT ∗ T is equivalent to (cid:20) Σ K Σ L (cid:21) = (cid:20) Σ K Σ L (cid:21) (cid:20) T ∗ T + T ∗ T T ∗ T + T ∗ T T ∗ T + T ∗ T T ∗ T + T ∗ T (cid:21) ,
10r equivalently Σ K = Σ K ( T ∗ T + T ∗ T ) + Σ L ( T ∗ T + T ∗ T ) , (3.11)Σ L = Σ K ( T ∗ T + T ∗ T ) + Σ L ( T ∗ T + T ∗ T ) . (3.12)Also, by Theorem 2.14 (i) we have that A = T T ∗ A , which in turn is equivalent toΣ K = ( T T ∗ + T T ∗ )Σ K (3.13)Σ L = ( T T ∗ + T T ∗ )Σ L (3.14)0 = ( T T ∗ + T T ∗ )Σ K (3.15)0 = ( T T ∗ + T T ∗ )Σ L (3.16)Multiplying (3.13) by K ∗ on right and (3.14) by L ∗ on right, adding and by using the identity KK ∗ + LL ∗ = I r we obtain T T ∗ + T T ∗ = I r . (3.17)Similarly, multiplying (3.15) on right by K ∗ and (3.16) by L ∗ on right and adding we have T T ∗ + T T ∗ = 0 . (3.18)Since R ( A ) = R ( T A ∗ ) implies AA † T A ∗ = T A ∗ , from Theorem 3.13 we obtain AA † T A ∗ = T A ∗ ⇔ (cid:20) T (Σ K ) ∗ + T (Σ L ) ∗
00 0 (cid:21) = (cid:20) T (Σ K ) ∗ + T (Σ L ) ∗ T (Σ K ) ∗ + T (Σ L ) ∗ (cid:21) , whence Σ KT ∗ + Σ LT ∗ = 0 . (3.19)Note that (3.11) can be rewritten asΣ K = Σ K ( T ∗ T + T ∗ T ) + Σ L ( T ∗ T + T ∗ T )= Σ KT ∗ T + Σ KT ∗ T + Σ LT ∗ T + Σ LT ∗ T = (Σ KT ∗ T + Σ LT ∗ ) T + (Σ KT ∗ T + Σ LT ∗ T ) . Thus, by (3.19) we have Σ K = Σ KT ∗ T + Σ LT ∗ T . (3.20)Similarly, from (3.11) and (3.19), we haveΣ L = Σ KT ∗ T + Σ LT ∗ T . (3.21)Multiplying (3.20) by K ∗ on right and (3.21) by L ∗ and adding and rearranging, we have I r = ( KT ∗ + LT ∗ )( T K ∗ + T L ∗ ) , (3.22)because Σ is nonsingular and KK ∗ + LL ∗ = I r .Now, we define the matrix D := Σ KT ∗ + Σ LT ∗ , which is nonsingular by (3.22) and thenonsingularity of Σ. By using (3.19), we have AT ∗ = U (cid:20) D
00 0 (cid:21) U ∗ . (3.23)11ince A = AT ∗ T from (3.17), (3.18) and (3.23), we have A = U (cid:20) D
00 0 (cid:21) U ∗ T, where T = U (cid:20) T T T T (cid:21) U ∗ such that T T ∗ + T T ∗ = I r and T T ∗ + T T ∗ = 0 hold.The proofs of (ii) ⇒ (iii) and (iii) ⇒ (i) are similar to (ii) ⇒ (iii) and (iii) ⇒ (i) of Theorem3.2. Finally, we suppose that any one of equivalent conditions (i)-(iii) holds. Then, CT T ∗ = U (cid:20) D
00 0 (cid:21) (cid:20) I r Z (cid:21) U ∗ = U (cid:20) D
00 0 (cid:21) U ∗ = C. This completes the proof.
Remark 3.15.
Note that the matrix
T T ∗ in (ii) ⇒ (iii) takes the form T T ∗ = U (cid:20) I r Z (cid:21) U ∗ , where Z = T T ∗ + T T ∗ is actually a hermitian matrix. The following theorem was proved by Pearl [14] and can be deduced from Theorem 3.14when T = I n . Corollary 3.16.
Let A ∈ C n × n and rank( A ) = r . Then A is EP if and only if there exists aunitary matrix U ∈ C n × n , and a nonsingular matrix D of order r such that A = U (cid:20) D
00 0 (cid:21) U ∗ . T -EP matrices As said earlier, for this section again we take all matrices as square matrices. We obtainseveral characterization of T -EP matrices when the partial isometry T satisfies some additionalconditions. More precisely, when T is either an orthogonal projector or a unitary matrix or anormal matrix.We have seen in Example 2.9 that a EP matrix is always relative EP respect to the identitymatrix of same order. This motivates the following question: When can an EP matrix be a T -EP matrix with respect to a nontrivial partial isometry T ?The following results show the relationship between EP and T -EP matrices. Theorem 4.1.
Let
A, T ∈ C n × n . If A is EP and AA ∗ is T -EP. Then A is T -EP.Proof. Since AA ∗ is T -EP, by definition we have R ( AA ∗ ) = R ( T ( AA ∗ ) ∗ T ) = R ( T AA ∗ T ) and N ( T ) ⊆ N ( AA ∗ ). Thus, N ( T ) ⊆ N ( AA ∗ ) = N ( A ∗ ) = N ( A ) , (4.1)where the last equality is due the fact that A is EP. Also, note that R ( A ) = R ( AA ∗ ) = R ( T AA ∗ T ) ⊆ R ( T A ) = T R ( A ) = T R ( A ∗ ) = R ( T A ∗ ) , and rank( T A ∗ ) ≤ rank( A ∗ ) = rank( A ). Therefore, R ( A ) = R ( T A ∗ ). In consequence, from(4.1) and Theorem 2.13 (iv) we have A is T -EP.12 emark 4.2. Notice that the condition AA ∗ is T -EP in above theorem can be replaced by AA † is T -EP. In fact, N ( AA ∗ ) = N ( AA † ) = N ( A † ) = N ( A ∗ ).Since every unitary matrix is also a partial isometry we have the following: Theorem 4.3.
Let
A, T ∈ C n × n . Suppose T is an unitary matrix. Then the following areequivalent: (i) A is T -EP; (ii) R ( A ) = R ( T A ∗ ) ; (iii) N ( A ∗ ) = N ( AT ∗ ) ; (iv) AT ∗ is EP; (v) T A ∗ is EP; (vi) T A † is EP; (vii) T A † A = AA † T .In particular, under any one of equivalent conditions (i)-(vii) we have ( a ) ( AT ) † = T ∗ A † , ( b ) ( T A ) † = A † T ∗ , and ( c ) ( T AT ) † = T ∗ A † T ∗ . (4.2) Proof. As T is unitary, the first three equivalences follow from Theorem 2.13. Similarly, equiv-alences (i) ⇔ (vi) ⇔ (vii) ⇔ (viii) follow from Theorem 3.10. Also, by Theorem 3.7 it is clear(i) ⇔ (vii) holds. Finally, the expressions in (4.2) can be easily proved by mean a directverification of the definition of Moore-Penrose inverse and by applying again the fact that T T ∗ = T ∗ T = I n .Recall that an involutory matrix is a nonsingular matrix that is its own inverse. Next,we present interesting characterizations of T -EP matrices when T is an involutory hermitianmatrix, that is, T − = T = T ∗ . Theorem 4.4.
Let
A, T ∈ C n × n . Suppose T is an involutory hermitian matrix. Then thefollowing are equivalent: (i) A is T -EP; (ii) R ( A ) = R ( T A ∗ ) ; (iii) N ( A ∗ ) = N ( AT ) ; (iv) AT is EP; (v) T A ∗ is EP; (vi) T A † is EP; (vii) T A † A = AA † T ; (viii) R ( A ∗ ) = R ( T A ) ; N ( A ) = N ( A ∗ T ) ; (x) A ∗ T is EP; (xi) T A is EP; (xii) A † T is EP; (xiii) A † AT = T AA † .Proof. As T satisfies T − = T = T ∗ , equivalences (i) to (xii) follow from Theorems 3.9 and 4.3.(iv) ⇔ (xiii). It is clear that ( AT ) † = T A † because T − = T = T ∗ . Moreover, T = I n . Thus, AT is EP if and only if AT ( AT ) † = ( AT ) † ATAT T A † = T A † ATAA † = T A † ATT AA † = A † AT.
Theorem 4.5.
Let
A, T ∈ C n × n . Suppose T is an orthogonal projector. Then the following areequivalent: (i) A is T -EP; (ii) A is EP and A = AT ; (iii) A ∗ is EP and A = AT ; (iv) A † is EP and A = AT .In particular, under any one of equivalent conditions (i)-(iv) we have that AT , T A , and also
T AT are all EP and T -EP.Proof. Note that if T is an orthogonal projector, A = AT ∗ T if and only if A = AT . Now, allequivalences follow as a direct application of Theorem 3.10.In order to prove the last affirmation we suppose (ii) holds. Clearly, A = AT . On the otherhand, by Theorem 2.14 (i) we know that A = T T ∗ A or equivalently A = T A , as T is anorthogonal projector. Consequently, A = AT = T A and so A = T AT . Therefore, part (ii)implies A is T -EP and therefore AT , T A , and also
T AT are all EP and T -EP. Lemma 4.6.
Let
A, T ∈ C n × n . Let T be a normal partial isometry. Then the following hold: (i) If A = T T ∗ A then ( T A ) † = A † T ∗ . In particular, A † = ( T A ) † T . (ii) If A = AT ∗ T then ( AT ) † = T ∗ A † . In particular, A † = T ( AT ) † . roof. (i) We check this by direct verification. Let X := A † T ∗ . Since A = T T ∗ A and T isnormal (i.e., T T ∗ = T ∗ T ) we have, T AXT A = T AA † T ∗ T A = T AA † T T ∗ A = T AA † A = T A,XT AX = A † T ∗ T AA † T ∗ = A † T T ∗ AA † T ∗ = A † AA † T ∗ = A † T ∗ = X, ( T AX ) ∗ = ( T AA † T ∗ ) ∗ = T ( AA † ) ∗ T ∗ = T AA † T ∗ = T AX, ( XT A ) ∗ = ( A † T ∗ T A ) ∗ = ( A † T T ∗ A ) ∗ = ( A † A ) ∗ = A † A = A † T T ∗ A = A † T ∗ T A = XT A.
By uniqueness of the Moore-Penrose inverse we have X = ( T A ) † .It remains to show the last affirmation of part (i). In fact, from Lemma 2.12 (i) and the factthat T is normal we obtain A † = A † T T ∗ = A † T ∗ T = ( T A ) † T .(ii) Can be proved similarly. Theorem 4.7.
Let
A, T ∈ C n × n . Let T be a normal partial isometry. If A is T -EP then theMoore-Penrose inverse of A is given by A † = ( T A ) † T = T ( AT ) † . Proof. As A is T -EP we know that A = AT ∗ T by definition. Also, from Theorem 2.14 (i) wehave A = T T ∗ A . Now, the expression of A † follows from Lemma 4.6. Theorem 4.8.
Let
A, T ∈ C n × n . Let T be an hermitian partial isometry. Then A is T -EP ifand only if T A is EP and A = AT = T A .Proof. Let A be T -EP. Then by definition of T -EP matrix and Theorem 2.14 (i), respectively,we have A = AT ∗ T and A = T T ∗ A which are equivalent to A = AT = T A as T is hermitian.In order to show that T A is EP is equivalent we will use the well-known characterization
T A ( T A ) † = ( T A ) † T A . In fact, first we note that T is a normal partial isometry. Thus, as A = T T ∗ A , Lemma 4.6 (i) implies ( T A ) † = A † T ∗ = A † T . Moreover, ( T A ) † T A = A † A . Also,Lemma 2.11 (iv) and Theorem 3.7 imply A † = T ∗ T A † = T A † and AA † T = T A † A , respectively.Therefore, T A ( T A ) † = T AA † T = T T A † A = A † A = ( T A ) † T A.
Conversely, let
T A be EP and A = T A = AT . Clearly, A = T T ∗ A and T is a normal partialisometry. Thus, by Lemma 4.6 (i) we have ( T A ) † = A † T ∗ = A † T . Therefore, T A ( T A ) † =( T A ) † T A is equivalent to
T AA † T = A † T A = A † A . Multiplying on left this equality, by T and using A = AT ∗ T = T A = AT implies AA † T = T A † A . Thus, Theorem 3.7 completes theproof. Theorem 4.9.
Let
A, T ∈ C n × n , and T a partial isometry such that A and T ∗ commute. Then A is T -EP if and only if A is EP and A = AT ∗ T. roof. Suppose A and T ∗ commute and A is T -EP. By definition, R ( A ) = R ( T A ∗ T ) . Since A and T ∗ commute, A ∗ and T commute. Therefore R ( A ) = R ( A ∗ T ) ⊆ R ( A ∗ ) . However,rank( A ) = rank( A ∗ ) implies R ( A ) = R ( A ∗ ) . Thus A is EP.Conversely, let A be EP and A = AT ∗ T. As A is EP, R ( A ) = R ( A ∗ ) . Thus, R ( T A ∗ ) = R ( AT ∗ ) ⊆ R ( A ∗ ) = R ( A ) . As A = AT ∗ T, we obtain rank( A ) = rank( AT ∗ T ) ≤ rank( AT ∗ ) = rank( T A ∗ ). So, R ( T A ∗ ) = R ( A ) . Now, by Theorem 2.13 (v), A is T -EP. T -EP matrices We now study the problem: When is sum of two T -EP matrices a T -EP matrix? This problemis not completely resolved even in case of EP matrices and we intend to explore it for T -EPmatrices in this final section. Theorem 5.1.
Let
A, B ∈ C m × n be T -EP such that A ∗ B + B ∗ A = 0 . Then A + B is T -EP.Proof. Let A and B be T -EP. As A ∗ B + B ∗ A = 0 we have T A ∗ BT ∗ + T B ∗ AT ∗ = 0, and so( AT ∗ + BT ∗ ) ∗ ( AT ∗ + BT ∗ ) = ( T A ∗ + T B ∗ )( AT ∗ + BT ∗ )= T A ∗ AT ∗ + T A ∗ BT ∗ + T B ∗ AT ∗ + T B ∗ BT ∗ = T A ∗ AT ∗ + T B ∗ BT ∗ . Thus, N ( AT ∗ + BT ∗ ) = N (( AT ∗ + BT ∗ ) ∗ ( AT ∗ + BT ∗ ))= N ( T A ∗ AT ∗ + T B ∗ BT ∗ )= N (cid:18)(cid:20) AT ∗ BT ∗ (cid:21) ∗ (cid:20) AT ∗ BT ∗ (cid:21)(cid:19) = N (cid:20) AT ∗ BT ∗ (cid:21) = N ( AT ∗ ) ∩ N ( BT ∗ ) . From Theorem 3.10 we know that AT ∗ is EP and A = AT ∗ T , and also BT ∗ is EP and B = BT ∗ T . Therefore, N ( AT ∗ + BT ∗ ) = N ( AT ∗ ) ∩ N ( BT ∗ )= N ( T A ∗ ) ∩ N ( T B ∗ ) ⊆ N ( T A ∗ + T B ∗ )= N ( T ( A + B ) ∗ )= N (( AT ∗ + BT ∗ ) ∗ ) , whence N ( AT ∗ + BT ∗ ) = N (( AT ∗ + BT ∗ ) ∗ ). Thus, ( A + B ) T ∗ is EP. Also, it is clear that A + B = ( A + B ) T ∗ T . Now, the result follows from Theorem 3.10. Corollary 5.2.
Let
A, B ∈ C m × n be T -EP such that A ∗ B = 0 . Then, A + B is T -EP. roof. It is a direct consequence from Theorem 5.1.
Corollary 5.3.
Let
A, B ∈ C m × n be T -EP such that BA ∗ = 0 . Then, A + B is T -EP.Proof. Firstly note that BA ∗ = 0 can be written as ( B ∗ ) ∗ A ∗ = 0. Also, by Theorem 3.9 weknow that A ∗ and B ∗ are T ∗ -EP. Thus, from Corollary 5.2 we have that B ∗ + A ∗ is T ∗ -EP.Thus, by applying again Theorem 3.9 we get A + B is T -EP.An interesting consequence of previous corollaries is the following theorem. Before we needtot define the concept of ∗ -orthogonality [8] of two matrices. Definition 5.4.
Let
A, B ∈ C m × n . We say A and B are ∗ -orthogonal in case A ∗ B = 0 and BA ∗ = 0 . It is easy to see that A and B are ∗ -orthogonal if and only if A ∗ and B ∗ are ∗ -orthogonal.Moreover, if A and B are both T -EP, then (i) A and B are ∗ -orthogonal implies AT ∗ and BT ∗ are ∗ -orthogonal and (ii) T A ∗ and T B ∗ are ∗ -orthogonal. Theorem 5.5.
Let
A, B ∈ C m × n be T -EP. If A and B are ∗ -orthogonal, then A + B is T -EP.Proof. Follows by Definition 5.4 and Corollary 5.2.In the following two results we present another sufficient conditions for a sum of two T -EPmatrices to be T -EP. Theorem 5.6.
Let
A, B ∈ C m × n be T -EP such that R ( A ) ∩ R ( B ) = { } . Then, A + B is T -EP.Proof. Since A and B are T -EP, by Theorem 3.10 we have AT ∗ is EP and A = AT ∗ T , and also BT ∗ is EP and B = BT ∗ T. We will show that ( A + B ) T ∗ is EP and ( A + B ) = ( A + B ) T ∗ T .Clearly, the second condition is true. On the other hand, note that N (cid:18)(cid:20) AT ∗ BT ∗ (cid:21)(cid:19) = N ( AT ∗ ) ∩ N ( BT ∗ ) ⊆ N ( AT ∗ + BT ∗ ) . (5.1)Also, we have R ( AT ∗ ) ∩ R ( BT ∗ ) ⊆ R ( A ) ∩ R ( B ) = { } . In consequence, as AT ∗ and BT ∗ areEP, also we have R ( T A ∗ ) ∩ R ( T B ∗ ) = { } . Thus,rank( AT ∗ + BT ∗ ) = rank( AT ∗ ) + rank( BT ∗ ) = rank (cid:20) AT ∗ BT ∗ (cid:21) . (5.2)Therefore, from (5.1) and (5.2) we obtain N ( AT ∗ + BT ∗ ) = N (( AT ∗ + BT ∗ ) ∗ ), i.e., ( A + B ) T ∗ is EP. Once again by Theorem 3.10, A + B is T -EP. Theorem 5.7.
Let
A, B ∈ C m × n be T -EP such that R ( A ∗ ) ∩ R ( B ∗ ) = { } . Then, A + B is T -EP.Proof. Let A and B be T -EP. From Theorem 3.9 we know that A ∗ and B ∗ are T ∗ -EP. Thus,Theorem 5.6 implies A ∗ + B ∗ is T ∗ -EP. Once again Theorem 3.9 we obtain that A + B is T -EP.We end this section with yet another sufficient conditions for a sum of two T -EP matricesto be T -EP. 17 heorem 5.8. Let
A, B ∈ C m × n be T -EP such that rank( T ( A + B ) ∗ T ) = rank( A + B ) and R ( A + B ) = R ( A ) + R ( B ) . Then A + B is T -EP.Proof. As A and B are T -EP, by definition we have R ( A ) = R ( T A ∗ T ) and A = AT ∗ T , andalso R ( B ) = R ( T B ∗ T ) and B = BT ∗ T . In consequence, from R ( A + B ) = R ( A ) + R ( B ) weobtain R ( T ( A + B ) ∗ T ) ⊆ R ( T A ∗ T ) + R ( T B ∗ T ) = R ( A ) + R ( B ) = R ( A + B ) . Thus, rank( T ( A + B ) ∗ T ) = rank( A + B ) implies R ( T ( A + B ) ∗ T ) = R ( A + B ) . Moreover, it iseasy to see ( A + B ) = ( A + B ) T ∗ T. Hence A + B is T -EP. Remark 5.9.
Note that if A and B are T -EP and ∗ -orthogonal then (i) rank( T ( A + B ) ∗ T ) =rank( A + B ) and (ii) R ( A + B ) = R ( A ) + R ( B ). In fact, (i) it follows from Corollary 5.5 andTheorem 2.14 (ii).(ii) It is a consequence of the fact that A ∗ B = 0 and BA ∗ = 0 imply R ( A ) ∩ R ( B ) = { } and R ( A ∗ ) ∩ R ( B ∗ ) = { } , respectively. Thus, rank( A + B ) = rank( A ) + rank( B ) and therefore R ( A + B ) = R ( A ) + R ( B ). References [1] Baksalary, O.M., Trenkler, G.: Characterizations of EP, normal and Hermitian matrices.Linear Multilinear Algebra , 299-304 (2008)[2] Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear transformations, SIAM,Philadelphia (2009)[3] Cheng, S., Tian, Y.: Two sets of new characterizations for normal and EP matrices. LinearAlgebra Appl. , 181-195 (2003)[4] Djordjevic, D.S.: Characterizations of normal, hyponormal and EP operators. J. Math.Anal. Appl. , 1181-1190 (2007)[5] Ferreyra, D.E., Levis, F.E., Thome, N.: Characterizations of k -commutative equalities forsome outer generalized inverses. Linear Multilinear Algebra (1), 177-192 (2020)[6] Hartwig, R.E., Spindelb¨ock, K.: Partial isometries, contractions and EP matrices. LinearMultilinear Algebra , 295-310 (1983)[7] Hartwig, R.E., Spindelb¨ock, K.: Matrices for which A ∗ and A † conmmute. Linear MultilinearAlgebra (3), 241-256 (1984)[8] Hestenes, M.R.: Relative Hermitian matrices. Pacific J. Math. , 224-245 (1961)[9] Malik, S.B., Rueda, L., Thome, N.: The class of m -EP and m -normal matrices. LinearMultilinear Algebra (11), 2119-2132 (2016)[10] Meenakshi, A.R., Indira, R.: On conjugate EP matrices. Kyungpook Math. J. , 67-72(1997) 1811] Meenakshi, A.R., Krishnamoorthy, S.: On k -EP matrices. Linear Algebra Appl. (1-3),219-232 (1998)[12] Mosic, D., Djordjevic, D. S., Koliha, J.J.: EP elements in rings. Linear Algebra Appl. ,527-535 (2009)[13] Pearl, M.H.: On normal and EP r matrices. Michigan Math. J. , 1-5 (1959)[14] Pearl, M.H.: On generalized inverses of matrices. Proc. Cambridge Phil. Sot. , 673-677(1966)[15] Penrose, R.: A generalized inverse for matrices. Math. Proc. Cambridge Philos. Soc. (3), 406-413 (1955)[16] Schwerdtfeger, S.: Introduction to linear algebra and the theory of matrices, P. Noordhoff,Groningen (1950)[17] Tian, Y., Wang, H.: Characterizations of EP matrices and weighted-EP matrices. LinearAlgebra Appl.434