Secondary range symmetric matrices

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.


Introduction
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. 4In 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. 6,7r an n Â n matrix A, the secondary transpose is related to transpose of the matrix by the relation A S ¼ VA T V. 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 A θ and is given by A θ ¼ 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 A D,θ9 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 Â n matrices with complex entries are denoted by C nÂn (R nÂn for matrices with real entries.Also, N A ð Þ represents the null space of the matrix A. The column space and rank of A are denoted by Then the secondary transpose (or secondary conjugate transpose A θ in the case of complex matrices) of A denoted by A S and is defined as Definition 2 (Ref.16) Let A∈ ℂ nÂn .Then the conjugate secondary transpose of A denoted by A θ and is defined as 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 AA S ¼ A S A.
Definition 4 (Ref.12) A † S is said to be secondary generalized inverse of A if and AA † S and A † S A are S À symmetric: 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.
Any further responses from the reviewers can be found at the end of the article Note that the matrix A is said to be secondary symmetric or S À symmetric if and only if A S ¼ A.
Theorem 0.1 (Ref.12) Given an m Â n matrix A. The following statements are equivalent.
(1) A has S -cancellation property (i.e., A s AX ¼ 0AX ¼ 0 and YAA s ¼ 0YA ¼ 0). ( Definition 5 (Ref.17) A matrix A∈ ℂ nÂn is said to be EP (or range symmetric 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 C n from 0 to n-1, be u tensor by G and it is defined as The Minkowski inner product on C n is defined by u, v ð Þ¼< 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∈ ℂ nÂn is said to be range symmetric in Minkowski space if and only if Here A þ represents the Minkowski adjoint given by A þ ¼ 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 (nÂn) if AXA ¼ A, XAX ¼ X and AX is S -symmetric.(XA is S -symmetric).
Example 1: is both secondary right normalized g-inverse and secondary left normalized inverse of A. The conditions AXA ¼ A, XAX ¼ X can be easily verified. Here Here X is also a left normalized g-inverse.
Definition 8 Consider A ∈ ℝ mÂn .The S -transpose of A is defined as . Therefore the matrix is not range symmetric, but it is secondary range symmetric.
Theorem 0.2 Let A ∈ ℝ nÂn .Then the following conditions are equivalent.
(1) A is secondary range symmetric where B and C are some nonsingular matrices This proves the equivalence of 1 ð Þ and 4 ð Þ.
A S ðBy property of secondary transposeÞ 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.
. Note that, here the secondary transpose A S of the matrix A coincides with A. Clearly A is secondary range symmetric (i.e., N A ð Þ¼N A S À Á : Here, A is secondary normal as well as . However, the matrix is not range symmetric in Minkowski's space since 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.
. Clearly B is not secondary range symmetric.
Observe that B is range symmetric in Minkowski space.Since, 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 A necessary condition for a matrix to be a S -EP (secondary range symmetric) is proved here.
. Thus A is secondary range symmetric.
A relation connecting range symmetric and secondary range symmetric matrices is given below: Theorem 0.4 Let A ∈ ℝ nÂn .Then any two of the following conditions imply the third one.
Hence A is range symmetric.Thus (1) holds.
For any square complex matrix A, there exists unique S-symmetric matrices such that 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∈ ℝ nÂn , A is secondary range symmetric if and only if
Since, both M and N are S-symmetric, they are secondary range symmetric.
. 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 Â
Theorem 0.6 For an n Â n matrix A, the following are equivalent: (1) A is secondary range symmetric and ρ A ð Þ¼ ρ A 2 À Á : (2) A † S exists and A † S is secondary range symmetric.
(3) There exists a symmetric idempotent matrix E such that AE ¼ EA and and A is secondary symmetric, by using Theorem 0.2 we have, ρ Hence by Thoerem 0.1, it follows that A † S exists, By Lemma 1 and Theorem 0.2, , by equivalence of condition ( 1) and ( 5) of Theorem 0.2, A † S is secondary range symmetric which implies that Since E is S -symmetric and idempotent, AA r is S-symmetric.Hence by definition 7, A n exists and AA n ¼ EE † S ¼ E which implies EA ¼ A. By hypothesis AE ¼ EA ¼ A. Therefore AA n ¼ A n A ¼ E. Thus both AA n and A n A are S -symmetric.By definition 2, A † S exists and E ¼ AA † S ¼ A † S A.

By taking secondary transpose on AE
. Thus (1) holds.Hence the theorem.
Corollary 1 Let A be n Â n secondary range symmetric matrix.Then exists A † S if and only if ρ A Proof 8 Since A is secondary range symmetric and ρ A ð Þ¼ρ A 2 À Á , 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.

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.

Is the work clearly and accurately presented and does it cite the current literature? Yes
Is the study design appropriate and is the work technically sound?Yes

Are sufficient details of methods and analysis provided to allow replication by others? 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 1.
The article requires a sufficient number of illustrations to explain the theory and definitions, which are currently missing.

2.
A new preliminary section should be included, containing all the necessary basic concepts to provide a strong foundation for the reader.

3.
Definition 7 does not adequately explain the entries of matrix A and needs to be revised for clarity.

4.
Definitions 7 and 8 should be illustrated through examples to enhance comprehension. 5.
The proofs, especially the proof of Theorem 1, are not written in an easily understandable manner and should be simplified for better readability.

6.
A conclusion and discussion section must be included and well explained to summarize the findings and implications of the study.

7.
The definition of Minkowski's space should be included, accompanied by examples to illustrate its application.

8.
The examples given at the beginning of page 5 are not properly explained and need to be elaborated for clarity.

9.
By addressing these points, the article can be significantly improved and provide a clearer, more comprehensive understanding of range symmetric matrices.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 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.

Is the work clearly and accurately presented and does it cite the current literature? No
Is the study design appropriate and is the work technically sound?Yes Are sufficient details of methods and analysis provided to allow replication by others?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

Are the conclusions drawn adequately supported by the results? Yes
Competing Interests: No competing interests were disclosed.
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.

6 :Definition 9
The Stranspose of A is given by A S ¼ A matrix A∈ ℝ nÂn secondary range symmetric if and only

©
2024 Manjhi P.This is an open access peer review report distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Pankaj Kumar Manjhi 1 Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India 2 Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India 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.

Comment 1 :
Are all the source data underlying the results available to ensure full reproducibility?PartlyAre the conclusions drawn adequately supported by the results?PartlyCompeting Interests: No competing interests were disclosed.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.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.

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.