
==== Front
F1000Res
F1000Res
F1000Research
2046-1402
F1000 Research Limited London, UK

10.12688/f1000research.144171.2
Research Article
Articles
Secondary range symmetric matrices
[version 2; peer review: 2 approved

Shenoy Divya Conceptualization Data Curation Formal Analysis Validation Writing – Original Draft Preparation https://orcid.org/0000-0001-7307-6688
a1
1 Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, Karnataka, 576104, India
a divya.shenoy@manipal.edu
No competing interests were disclosed.

11 9 2024
2024
13 11220 6 2024
Copyright: © 2024 Shenoy D
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The concept of secondary range symmetric matrices is introduced here. Some characterizations as well as the equivalent conditions for a range symmetric matrix to be secondary range symmetric matrix is given. The idea of range symmetric matrices, range symmetric matrices over Minkowski space and secondary range symmetric matrices are different, and is depicted with the help of suitable examples. Finally, a necessary and sufficient condition for a secondary range symmetric matrix to have a secondary generalized inverse has been obtained.

Generalized inverses
Secondary generalized inverses
Secondary transpose
EP matrices
The author(s) declared that no grants were involved in supporting this work.Revised Amendments from Version 1

Version 2 of the article incorporates the valuable suggestions from the reviewers. Changes include language correction, a preliminary section highlighting basic definitions, examples elaborating the definitions, etc. Additionally, a few references have been added, updating the bibliography.
==== Body
pmcIntroduction

The theory of symmetric matrices as well as range symmetric matrices are well known in literature. A matrix is said to be EP (or range symmetric), whenever the range space of the matrix is equal to the range space of its conjugate transpose. In other words, matrix is EP whenever its null space is same as that of the null space of its conjugate transpose. Ballantine 1 has studied about the product of two EP matrices of specific rank to be again an EP matrix. In 2 new characterizations of EP matrices are given. Also, weighted EP matrix is defined and characterized. Meenakshi 3 extended the concept of range symmetric matrices over Minkowski space. In 2014, the same author defined range symmetric matrices in indefinite inner product space. 4 In Ref. 5, the concept of EP matrices to bounded operator with closed range is defined on a Hilbert space. For more characterizations of EP and hypo EP operators one can refer. 6 , 7

For an n×n matrix A , the secondary transpose is related to transpose of the matrix by the relation AS=VATV . Here, the matrix V has non zero unitary entries only on the secondary diagonal. For a matrix A with complex entries, the secondary transpose will be renamed as secondary conjugate transpose Aθ and is given by Aθ=VA∗V .

The concept of secondary conjugate transpose is gaining importance in recent years. Shenoy 8 has defined Outer Theta inverse by combining the outer inverse and secondary transpose of a matrix A . Drazin-Theta matix AD,θ 9 is a new class of generalized inverse introduced for a square matrix of index m. One can refer 10 for the extension of these inverses over rectangular matrices. R. Vijayakumar 11 introduced the concept of secondary generalized inverse with the help of secondary transpose of a matrix. This concept is similar to Moore Penrose inverse. But unlike Moore penrose inverse, existence of s-g inverse is not assured in general. In Ref. 12, a necessary and sufficient conditions of existence of s-g invese is given. In the same article, a few characterizations and a determinantal formula for s-g inverse also has been discussed. In 2009, Krishnamoorthy and Vijayakumar 13 has defined the concept of S - normal matrices with the help of secondary transopse for a class of complex square matrices. Jayashree 14 has defined secondary k - range symmetric fuzzy matrices. Its relation with S - range symmetric fuzzy matrices, k - range fuzzy symmetric matrices and EP matrices are defined.

In this article, we define secondary range symmetric matrices. Several equivalent conditions for a matrix to be secondary range symmetric, is obtained here. Also, the existence of secondary generalized inverse of a secondary range symmetric matrix is discussed.

Below are some useful deifnitions and results related to secondary conjugate transpose.

Preliminaries

The set of all n×n matrices with complex entries are denoted by Cn×n ( Rn×n for matrices with real entries. Also, NA represents the null space of the matrix A. The column space and rank of A are denoted by CA,ρA respectively. Definition 1 (Ref. 15) Let A∈ℝn×n . Then the secondary transpose (or secondary conjugate transpose Aθ in the case of complex matrices) of A denoted by AS and is defined as AS=bij , where bij=an−j+1,n−i+1 , i,j=1,2,…n .

Definition 2 (Ref. 16) Let A∈ℂn×n . Then the conjugate secondary transpose of A denoted by Aθ and is defined as Aθ=A¯s=cij where cij=a¯n−j+1,n−i+1 .

The secondary transpose of a matrix is defined by reflecting the entries through sendary diagonal. Definition 3 (Ref. 16) A matrix is said to be secondary normal (S - normal) if AAS=ASA .

Definition 4 (Ref. 12) A†S is said to be secondary generalized inverse of A if AA†SA=AA†SAA†S=A†SandAA†SandA†SAareS−symmetric.

Note that the matrix A is said to be secondary symmetric or S−symmetric if and only if AS=A .

Theorem 0.1 (Ref. 12) Given an m×n matrix A . The following statements are equivalent. (1) A has S - cancellation property (i.e., AsAX=0AX=0 and YAAs=0YA=0 ).

(2) ρAsA=ρAAs=ρA

(3) A†S exists.

Definition 5 (Ref. 17) A matrix A∈ℂn×n is said to be EP (or range symmetric) if NA=NA∗ .

Meenakshi 3 has defined EP in Minkowski space and has given equivalent conditions for a matrix to be range symmetric.

Let the components of a complex vector Cn from 0 to n-1, be u=u0u1u2…un−1. Let us denote the Minkowski metric tensor by G and it is defined as G=u0−u1−u2…−un−1 . Now, the Minkowski metric matrix is G=100−In−1 . The Minkowski inner product on Cn is defined by uv=<u,Gv> where <.,.> is the Hilbert inner product. A space with Minkowski inner product is defined as Minkowski space. The idea of Minkowski space arised when Xing 18 tried to study the optical devices described by the Mueller matrix which may not have a singluar value decomposition. The problem was solved by Renardy 19 by defining Minkowski space and obtained the singular value decomposition of Mueller matrix over the Minkowski space.

Definition 6 (Ref. 3) A matrix A∈ℂn×n is said to be range symmetric in Minkowski space if and only if NA=NA+ .

Here A+ represents the Minkowski adjoint given by A+=GA∗G where G is the Minkowski metric tensor.

Results

In this section, we define secondary range symmetric matrices which is analogous to that of range symmetric matrices. Some equivalent conditions for a matrix to be range symmetric is also given here. Definition 7 X is secondary right (left) normalized g inverse of A where A ∈ R (n×n) if AXA=A , XAX=X and AX is S - symmetric. ( XA is S - symmetric).

Example 1:

Let A=112112. Note that the matrix X=−1107/5−4/51/5 is both secondary right normalized g-inverse and secondary left normalized inverse of A. The conditions AXA=A,XAX=X can be easily verified.

Also AX=2/51/51/5−3/56/51/59/5−3/52/5=AXS . Therefore X is secondary right normalized g-inverse of A.

Here XA=1001=XAS. Here X is also a left normalized g-inverse.

Definition 8 Consider A∈ℝm×n . The S - transpose of A is defined as As=bij where bij=am−j+1,n−i+1 , 1≤i≤n,1≤j≤m .

Example 2:

Consider a matrix A=123456. The S – transpose of A is given by AS=635241 .

Definition 9 A matrix A∈ℝn×n secondary range symmetric if and only if NA=NAS

Example 3:

Let A=1001. Here NA≠NA∗. But NA=NAS . Therefore the matrix is not range symmetric, but it is secondary range symmetric.

Theorem 0.2 Let A∈ℝn×n . Then the following conditions are equivalent. (1) A is secondary range symmetric

(2) VA is EP

(3) AV is EP (V is a permutation matrix with ‘1’ in the secondary diagonal)

(4) NA∗=NAV

(5) CA=CAS

(6) AS=BA=AC where B and C are some nonsingular matrices

(7) CA∗=CVA

(8) CA∗⊕NA=ℂn

(9) CA⊕NA∗=ℂn

Proof 1 A is secondary range symmetric⇔NA=NAS⇔NVA=NASV (sinceV2=I)⇔VA is EP⇔VVAVSis EP⇔AV is EP.

Hence 1⇔2⇔3 holds true.

1⇔4 A is secondary range symmetric⇔NA=NAS⇔NA=NVA∗V (by definition of secondary transpose)⇔NA=NA∗V⇔A∗V=A∗VA1A⇔A∗=A∗VA1AV⇔A∗=A∗AV1AV⇔NA∗=NAV

This proves the equivalence of 1 and 4 .

3⇔5 AV is range symmetric⇔CAV=CAV∗⇔CA=CVA∗ (SinceV*=V)⇔CA=CVA∗V⇔CA=CAS

Hence 3 and 5 are equivalent.

2⇔6 VAis range symmetric⇔VA∗=VAPfor some nonsingularn×nmatrixP⇔A∗V=VAP⇔VA∗V=AP⇔AS=AP⇔A=APS=PSAS (By property of secondary transpose)⇔AS=PS−1A⇔AS=KAwhereK=PS−1. (Using(A†S)†S=A)

Hence 2⇔6 .

5⇔7 CA=CAS⇔CA=CVASV⇔CA=CVAS⇔VA∗=AA1VA∗ (By[14])⇔A∗=VAA1VA∗⇔A∗=VAVA1A∗⇔CA∗=CVA

Thus equivalence of 5 and 7 are proved.

2 ⇔ 8 . VA is range symmetric⇔ℂn=CVA⊕NGA=CVA∗⊕NA=CA∗V⊕NA=CA∗⊕NA

Thus equivalence of 2 and 8 are proved.

3⇔9 AV is range symmetric⇔ℂn=CAV⊕NAV=ℂn=CAV⊕NAV∗=CA⊕NA∗

Thus the equivalence of 3 and 9 is proved.

From the following example, it is clear that EP matrices in Minkowski space defined by Meenakshi 3 and secondary range symmetric matrices are two different concepts.

Consider a matrix A=−111−1 where AS=−111−1 . Note that, here the secondary transpose A S of the matrix A coincides with A. Clearly A is secondary range symmetric (i.e., NA=NAS. Here, A is secondary normal as well as ρA=ρAAS . However, the matrix is not range symmetric in Minkowski’s space since A+=GA∗G=−1−1−1−1 . It is clear that NA≠NA+ .

In this example, the matrix A is secondary range symmetric, but not range symmetric in Minkowski space.

In the following example, B is range symmetric in Minkowski space. But it is not secondary range symmetric.

Let B=1−11−1 and BS=−1−111 . Clearly B is not secondary range symmetric.

Observe that B is range symmetric in Minkowski space. Since, B+=GB∗G=1−11−1 so that NB=NB+ .

These examples shows that secondary range symmetric matrices and range symmetric matrices in Minkowski inverse are two different matrices even though the proof techniques adopted here are similar.

Note that AS=VA∗V=VGA+GV .

A necessary condition for a matrix to be a S - EP (secondary range symmetric) is proved here. Theorem 0.3 Let A∈ℝn×n . If A is secondary normal and ρA=ρAAS , then A is secondary range symmetric.

Proof 2 Since A is secondary normal, AAS=ASA . Hence ρA=ρAAS=ρASA=ρAS which implies NA=NAAS=NASA=NAS . Thus A is secondary range symmetric.

A relation connecting range symmetric and secondary range symmetric matrices is given below: Theorem 0.4 Let A∈ℝn×n . Then any two of the following conditions imply the third one. (1) A is EP.

(2) A is secondary EP.

(3) CA=CVA .

Proof 3 1,2⇒3

Since A is EP, CA=CA∗ . By condition (7) of Theorem 0.2 , CA∗=CVA as A is secondary range symmetric. Hence A is secondary EP ⇔ CA=CVA .

(1), (3) ⇒ (2)

Since A is EP, CA=CA∗. Also, from (3) we have CA=CVA, which gives CA∗=CVA. Hence, by Theorem 0.2 , A is secondary range symmetric.

2,3⇒1

Since A is S - EP, by condition (2) of Theorem 0.2 , VA is range symmetric. Hence CVA=CVA∗=CA∗V=CA∗ . Also, By (3), CA=CVA from which it follows that CA=CA∗ . Hence A is range symmetric. Thus (1) holds.

For any square complex matrix A , there exists unique S - symmetric matrices such that A=M+iN where M=12A+AS and N=12iA−AS . In the following theorem, an equivalent condition for a matrix A to be secondary range symmetric is obtained interms of M , the S -symmetric part of A . Theorem 0.5 For A∈ℝn×n , A is secondary range symmetric if and only if NA⊆NM where M is the S - symmetric part of A .

Proof 4 If A is secondary range symmetric, then NA=NAS . For x∈NA , Ax=0 and ASx=0 . Hence Mx=0 . Thus, NA⊆NM , then Ax=0⇒Mx=0 and hence Nx=0 . Therefore NA⊆NN . Thus NA⊆NM∩NN . Since, both M and N are S -symmetric, they are secondary range symmetric. NM=NMS=NVM∗V=NM∗V

and NN=NNS=NVN∗V=NN∗V

Now, NA⊆NM∩NN=NM∗V∩NN∗V⊆NM∗+iN∗V=NA∗V and ρA=ρA∗=ρA∗V . Therefore, NA=NA∗V=NVA∗V=NAS . Thus A is secondary range symmetric.

We shall discuss the existence of secondary generalized inverse inverse of a secondary range symmetric matrix. First, we shall prove certain lemmas, to simplify the proof of the main result. Lemma 1 For an m×n matrix A , if A†S exists, then CA†S=CAS .

Proof 5 If A†S exists, then A†SA=A†SAS=ASAA†SSA†SS and A†S=A†SAA†S=ASA†SSA†S⇒CA†S⊆CAS.

Further, ρA†S=ρA=ρAS . Thus, CA†S=CAS .

Lemma 2 For an m×n matrix A , if A†S exists, then CAA†S is the projection on CA†S and A†SA is the projection on CA†S .

Proof 6 x∈CA if and only if x=Ay=AA†SAy=AA†Sx . By definition 2 , AA†S being S - symmetric, idempotent is the projection on CA . Similarly, x∈CA†S if and only if x=A†SAA†Sy=A†SAx and A†SA is S - symmetric and idempotent. Hence A†SA is the projection on CA†S .

Theorem 0.6 For an n×n matrix A , the following are equivalent: (1) A is secondary range symmetric and ρA=ρA2.

(2) A†S exists and A†S is secondary range symmetric.

(3) There exists a symmetric idempotent matrix E such that AE=EA and CA=CE.

Proof 7 1⇒2 . Since ρA=ρA2 and A is secondary symmetric, by using Theorem 0.2 we have, ρASA=ρBA2=ρA2=ρA and ρAAS=ρA2C=ρA2=ρA . Thus ρA=ρAAS=ρASA . Hence by Thoerem 0.1 , it follows that A†S exists, By Lemma 1 and Theorem 0.2 , CA†S=CAS=CA=CASS=CA†SS . Hence A†S is range symmetric. Thus (2) holds.

2⇒3 . Since A†S exists, by Lemma 1, CA†S=CAS , by equivalence of condition (1) and (5) of Theorem 0.2, A†S is secondary range symmetric which implies that CA†S=CA†SS . Hence CAS=CA†SS . By Lemma 2, it follows that ASAS†S=A†SSAS , hence A†SAS=AA†SS . By definition 2, A†SA=AA†S=E , is S - symmetric, idempotent and AE=EA=A ; hence CA⊆CE and ρE=ρAA†S=ρA , which implies CA=CE . Thus 3 holds.

3⇒1 . Since, E is S - symmetric and idempotent, ES=E=E2 , by lemma 1.1, E†S exists and E†S=E implies E is the projection on CA . For all reflexive g-inverses Ar of A , AAr=EE†S=E . Since E is S - symmetric and idempotent, AAr is S - symmetric. Hence by definition 7, An exists and AAn=EE†S=E which implies EA=A . By hypothesis AE=EA=A . Therefore AAn=AnA=E . Thus both AAn and AnA are S - symmetric. By definition 2, A†S exists and E=AA†S=A†SA . By taking secondary transpose on AE=EA=A , we get EAS=ASE=AS . CAS⊆CE=CA and ρAS=ρA . Therefore CA=CAS . By theorem 0.2, A is secondary ramge symmetric. ρAAS≥ρAASA†SS=ρAA†SS , ρAE=ρA≥ρAAS . Thus ρA=ρAAS=ρA2C=ρA2 . Thus (1) holds. Hence the theorem.

Corollary 1 Let A be n×n secondary range symmetric matrix. Then exists A†S if and only if ρA=ρA2 .

Proof 8 Since A is secondary range symmetric and ρA=ρA2 , the existence of A†S follows from equivalence of (1) and (2) of Thereom 0.6. Conversly, if A is secondary range symmetric, and A†S exists, then by equivalence of (2) and (3) of Theorem 0.1 ρA=ρAAS=ρASA and by Theorem 0.2 , AS=AC . Hence ρA=ρAAC=ρA2 .

Conclusion

In this article we defined and characterized the concept of secondary range symmetric matrices. The Moore Penrose inverse exists for any matrix. But, in the case of secondary generalized inverse this is not true. Here, we obtained a necessary condition for a secondary range symmetric matrix to have an s-g inverse. In fact this condition holds true for the existence of secondary generalized inverse for any matrix.

As an extension of this work, the sum of range symmetric matrices are discussed in Ref. 20. One can think of defining weighted secondary EP matrices and its characterizations. Also, extending secondary range symmetric matrix to indefinite inner product spaces will open up a new area of research.

Data availability

No data are associated with this article.

10.5256/f1000research.157925.r261048
Reviewer response for version 1
Atteya Mehsin Jabel 1Referee https://orcid.org/0000-0001-7380-6951

1 Department of Mathematics, Al- Mustansiriyah University, Falastin St, Baghdad, Iraq
8 6 2024 Copyright: © 2024 Atteya MJ
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove
Dear Sir.,

My Report:

1. The Definitions 7, 8 and 9 need to provide examples.

2. In the proof of Theorem 0.4, where the prove of the case (1) and (3) implies to (2).

3. in the proof of Corollary 1, the author used Theorem 0.1 while Theorem 0.1 concerning    an m×n  matrix A.

4. Again in Corollary 0.1, the author depended on Theorems (0.1 and 0.6) without mention the branches.

Similarly for the prove of Theorems(0.3 and 0.6) where the author used Theorems (0.2 and 0.1).

5. The branches of Theorem (0.2) wrote by Roman numbers system while in the proof used   Arabic numbers system.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

The paper tried to supply new results with avoid any mistake.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Shenoy Divya Mathematics, Manipal Academy of Higher Education, Manipal, Karnataka, India

19 6 2024 Comments and Responses:

Comment 1: The Definitions 7, 8 and 9 need to provide examples.

Response:  Definitions 7,8 and 9 are illustrated with examples- 1, 2 and 3 respectively.

Comment 2: In the proof of Theorem 0.4, where the prove of the case (1) and (3) implies to (2).

Response: The omission error has been corrected in the revised article. A proof for (1) and (3) implies (2) is added in the new version.

Comment 3: In the proof of Corollary 1, the author used Theorem 0.1 while Theorem 0.1 concerning   an m×n  matrix A.

Response: Theorem 0.1 holds for a matrix of any order [m×n  as well as n×n ].

For generalization, the authors [in reference: V. Savitha, D. P. Shenoy, K. Umashankar and R. B. Bapat, Secondary transpose of a matrix and generalized inverses, Journal of Algebra and its applications, 23(3), 2450052, 2024, doi:10.1142/S021949882450052X.] have considered rectangular matrices. Similar to Moore-Penrose inverse, in this context, the conditions for rectangular matrices holds true for square matrices also.

Comment 4: Again in Corollary 0.1, the author depended on Theorems (0.1 and 0.6) without mention the branches. Similarly for the prove of Theorems(0.3 and 0.6) where the author used Theorems (0.2 and 0.1).

Response: The appropriate branches are mentioned in the revised version. 

Comment 5: The branches of Theorem (0.2) wrote by Roman numbers system while in the proof used   Arabic numbers system.

Response: The typological errors have been corrected.

10.5256/f1000research.157925.r274406
Reviewer response for version 1
Manjhi Pankaj Kumar 1Referee
1 Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India
8 6 2024 Copyright: © 2024 Manjhi PK
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove-with-reservations
I am very delighted to see the scholarly interest in the study of range symmetric matrices. Here are my suggestions for improving the article: Some definitions (such as definition 1) are not clearly written and need to be clarified for better understanding.

The article requires a sufficient number of illustrations to explain the theory and definitions, which are currently missing.

A new preliminary section should be included, containing all the necessary basic concepts to provide a strong foundation for the reader.

Definition 7 does not adequately explain the entries of matrix A and needs to be revised for clarity.

Definitions 7 and 8 should be illustrated through examples to enhance comprehension.

The proofs, especially the proof of Theorem 1, are not written in an easily understandable manner and should be simplified for better readability.

A conclusion and discussion section must be included and well explained to summarize the findings and implications of the study.

The definition of Minkowski’s space should be included, accompanied by examples to illustrate its application.

The examples given at the beginning of page 5 are not properly explained and need to be elaborated for clarity.

By addressing these points, the article can be significantly improved and provide a clearer, more comprehensive understanding of range symmetric matrices.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Yes

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Combinatorial matrices, Discrete Mathematics, computer science

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Shenoy Divya Mathematics, Manipal Academy of Higher Education, Manipal, Karnataka, India

19 6 2024 Comments and Responses:

Comment 1: Some definitions (such as definition 1) are not clearly written and need to be clarified for better understanding.

Response: Definition 1 (of the secondary transpose of a rectangular matrix) has been quoted from: A. Lee, Secondary symmetric, Skew symmetric and orthogonal matrices, Period. Math. Hungar 7(1), pp. 63-70, 1976.

In the revised article, it is elaborated and an example [example-2, results] has been added for better understanding of the concept.

Comment 2: The article requires a sufficient number of illustrations to explain the theory and definitions, which are currently missing.

Response: As per the suggestion, the article is modified providing sufficient examples. On page 3, the definitions are explained with examples [example 1, 2 and 3]. Also, a clearer explanation is given for the example given after the proof of theorem 0.1.

Comment 3: A new preliminary section should be included, containing all the necessary basic concepts to provide a strong foundation for the reader.

Response: A preliminary section [ section 2] is included with all necessary basic concepts.

Comment 4: Definition 7 does not adequately explain the entries of matrix A and needs to be revised for clarity.

Response: The definition (7) has been revised. The entries of the matrix are from a real field. The explanation has been provided in the text.

Comment 5: Definitions 7 and 8 should be illustrated through examples to enhance comprehension.

Response: As per the suggestion, definitions 7 and 8 are explained with examples.

Comment 6: The proofs, especially the proof of Theorem 1, are not written in an easily understandable manner and should be simplified for better readability.

Response: Explanation for the steps for the proof of Theorem -1, has been provided within parenthesis.

Comment 7: A conclusion and discussion section must be included and well explained to summarize the findings and implications of the study.

Response:  The conclusion section has been modified to include discussion, and scope for future work.

Comment 8: The definition of Minkowski’s space should be included, accompanied by examples to illustrate its application

Response: As per the suggestion, the definition of Minkowski space, with application and reference is included in the preliminary section.

Comment 9: The examples given at the beginning of page 5 are not properly explained and need to be elaborated for clarity.

Response: The example is elaborated for better understanding.

10.5256/f1000research.157925.r252908
Reviewer response for version 1
Johnson P Sam 1Referee https://orcid.org/0000-0003-3461-5380

1 National Institute of Technology Karnataka, Surathkal, Surathkal, Karnataka,, India
28 3 2024 Copyright: © 2024 Johnson PS
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Version 1recommendationapprove
The paper is well written.  It discusses secondary range symmetric matrices.  It should incorporate the following suggestions / corrections :

Minor points :

(i) First line of abstract should be "is" introduced.

(ii) No uniformity of symbol "s" in many places : s-symmetric, s-transpose ; capital letter or a small letter

(iii) Corollary 1 in page 6 should be rephrased as : Let A be an nxn secondary range symmetric matrix.

Major point:

(iv) The following recent references may be added.

Is the work clearly and accurately presented and does it cite the current literature?

No

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Functional Analysis

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Shenoy Divya Mathematics, Manipal Academy of Higher Education, Manipal, Karnataka, India

31 3 2024 I thank reviewer 1 for the critical comments.

The suggested corrections (typo errors) and the inclusion of latest references in the manuscript and their corresponding information in the body of the manuscript will be incorporated in version 1, which will be uploaded after receiving comments from the second reviewer.

Shenoy Divya Mathematics, Manipal Academy of Higher Education, Manipal, Karnataka, India

19 6 2024 Comments and Responses

A. Response to Minor points :

Comment (i): First line of abstract should be "is" introduced.

Response: The correction has been incorporated.

Comment (ii): No uniformity of symbol "s" in many places: s-symmetric, s-transpose; capital letter or a small letter.

Response: Uniformity has been maintained in the new version. Capital letter is used for all instances, S – symmetric, S – transpose etc.

Comment (iii): Corollary 1 in page 6 should be rephrased as: Let A be an nxn secondary range symmetric matrix.

Response: The suggested correction has been incorporated.

Major point:

Comment (iv): Regarding the addition of recent references on a) product and factorization of hypo-EP operators and b) Closed EP and hypo-EP operators on Hilbert spaces. 

Response: The latest references have been added. The relevant text is added in the article, with appropriate citation.

Competing interests: No competing interests were disclosed.

Competing interests: There are no competing interests to declare

Competing interests: No competing interests were disclosed.

Competing interests: There are no competing interests to declare

Competing interests: No competing interests were disclosed.

Competing interests: No competing interests.

Competing interests: There are no competing interest to declare.
==== Refs
References

1 Ballantine CS : Products of EP matrices. Linear Algebra Appl. 1975;12 (3 ):257–267. 10.1016/0024-3795(75)90048-8
2 Tian Y Wang H : Characterizations of EP matrices and Weighted EP matrices. Linear Algebra Appl. 2011;434 (5 ):1295–1318. 10.1016/j.laa.2010.11.014
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