Secondary range symmetric matrices

Divya Shenoy

doi:10.12688/f1000research.144171.2

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Research Article

Revised

Secondary range symmetric matrices

[version 2; peer review: 3 approved]

Divya Shenoy

PUBLISHED 11 Sep 2024

Author details Author details

Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, Karnataka, 576104, India

Divya Shenoy
Roles: Conceptualization, Data Curation, Formal Analysis, Validation, Writing – Original Draft Preparation

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Manipal Academy of Higher Education gateway.

Abstract

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.

Keywords

Generalized inverses, Secondary generalized inverses, Secondary transpose, EP matrices

Corresponding author: Divya Shenoy

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2024 Shenoy D. This is an open access article 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.

How to cite: Shenoy D. Secondary range symmetric matrices [version 2; peer review: 3 approved]. F1000Research 2024, 13:112 (https://doi.org/10.12688/f1000research.144171.2) First published: 19 Feb 2024, 13:112 (https://doi.org/10.12688/f1000research.144171.1) Latest published: 11 Sep 2024, 13:112 (https://doi.org/10.12688/f1000research.144171.2)

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.

See the author's detailed response to the review by P Sam Johnson
See the author's detailed response to the review by Mehsin Jabel Atteya
See the author's detailed response to the review by Pankaj Kumar Manjhi

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. Ballantine¹ has studied about the product of two EP matrices of specific rank to be again an EP matrix. In² new characterizations of EP matrices are given. Also, weighted EP matrix is defined and characterized. Meenakshi³ 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 $n \times n$ matrix $A$ , the secondary transpose is related to transpose of the matrix by the relation $A^{S} = {VA}^{T} V$ . 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 $A^{θ} = {VA}^{*} V$ .

The concept of secondary conjugate transpose is gaining importance in recent years. Shenoy⁸ has defined Outer Theta inverse by combining the outer inverse and secondary transpose of a matrix $A$ . Drazin-Theta matix $A^{D, θ}$ ⁹ is a new class of generalized inverse introduced for a square matrix of index m. One can refer¹⁰ for the extension of these inverses over rectangular matrices. R. Vijayakumar¹¹ 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 Vijayakumar¹³ has defined the concept of S - normal matrices with the help of secondary transopse for a class of complex square matrices. Jayashree¹⁴ 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 $n \times n$ matrices with complex entries are denoted by $C^{n \times n}$ ( $R^{n \times n}$ for matrices with real entries. Also, $N (A)$ represents the null space of the matrix A. The column space and rank of A are denoted by $C (A), ρ (A)$ respectively.

Definition 1

(Ref. 15) Let $A \in ℝ^{n \times n}$ . 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 $A^{S} = (b_{ij})$ , where $b_{ij} = a_{n - j + 1, n - i + 1}$ , $i, j = 1, 2, \dots n$ .

Definition 2

(Ref. 16) Let $A \in ℂ^{n \times n}$ . Then the conjugate secondary transpose of $A$ denoted by $A^{θ}$ and is defined as $A^{θ} = {\bar{A}}^{s} = (c_{ij})$ where $c_{ij} = {\bar{a}}_{n - j + 1, n - i + 1}$ .

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 ${AA}^{S} = A^{S} A$ .

Definition 4

(Ref. 12) $A^{†_{S}}$ is said to be secondary generalized inverse of $A$ if

{AA}^{†_{S}} A = A A^{†_{S}} {AA}^{†_{S}} = A^{†_{S}} and {AA}^{†_{S}} and A^{†_{S}} A are S - symmetric .

Note that the matrix A is said to be secondary symmetric or $S - symmetric$ if and only if $A^{S} = A$ .

Theorem 0.1

(Ref. 12) Given an $m \times n$ matrix $A$ . The following statements are equivalent.

(1) $A$ has S - cancellation property (i.e., $A^{s} AX = 0 AX = 0$ and ${YAA}^{s} = 0 YA = 0$ ).
(2) $ρ (A^{s} A) = ρ ({AA}^{s}) = ρ (A)$
(3) $A^{†_{S}}$ exists.

Definition 5

(Ref. 17) A matrix $A \in ℂ^{n \times n}$ is said to be EP (or range symmetric) if $N (A) = N (A^{*})$ .

Meenakshi³ 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 $u = (u_{0} u_{1} u_{2} \dots u_{n - 1}) .$ Let us denote the Minkowski metric tensor by $G$ and it is defined as $G = (u_{0} - u_{1} - u_{2} \dots - u_{n - 1})$ . Now, the Minkowski metric matrix is $G = (\begin{matrix} 1 & 0 \\ 0 & - I_{n - 1} \end{matrix})$ . The Minkowski inner product on $C^{n}$ is defined by $(u, v) = < u, Gv >$ where $< ., . >$ is the Hilbert inner product. A space with Minkowski inner product is defined as Minkowski space. The idea of Minkowski space arised when Xing¹⁸ tried to study the optical devices described by the Mueller matrix which may not have a singluar value decomposition. The problem was solved by Renardy¹⁹ by defining Minkowski space and obtained the singular value decomposition of Mueller matrix over the Minkowski space.

Definition 6

(Ref. 3) A matrix $A \in ℂ^{n \times n}$ is said to be range symmetric in Minkowski space if and only if $N (A) = N (A^{+})$ .

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 $A = (\begin{matrix} 1 & 1 \\ 2 & 1 \\ 1 & 2 \end{matrix}) .$ Note that the matrix $X = (\begin{matrix} - 1 & 1 & 0 \\ 7 / 5 & - 4 / 5 & 1 / 5 \end{matrix})$ 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 $AX = (\begin{matrix} 2 / 5 & 1 / 5 & 1 / 5 \\ - 3 / 5 & 6 / 5 & 1 / 5 \\ 9 / 5 & - 3 / 5 & 2 / 5 \end{matrix}) = {(AX)}^{S}$ . Therefore X is secondary right normalized g-inverse of A.

Here $XA = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) = {(XA)}^{S} .$ Here X is also a left normalized g-inverse.

Definition 8

Consider $A \in ℝ^{m \times n}$ . The S - transpose of $A$ is defined as $A^{s} = (b_{ij})$ where $b_{ij} = a_{m - j + 1, n - i + 1}$ , $1 \leq i \leq n, 1 \leq j \leq m$ .

Example 2:

Consider a matrix $A = (\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}) .$ The S – transpose of A is given by $A^{S} = (\begin{matrix} 6 & 3 \\ 5 & 2 \\ 4 & 1 \end{matrix})$ .

Definition 9

A matrix $A \in ℝ^{n \times n}$ secondary range symmetric if and only if $N (A) = N (A^{S})$

Example 3:

Let $A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .$ Here $N (A) \neq N (A^{*}) .$ But $N (A) = N (A^{S})$ . Therefore the matrix is not range symmetric, but it is secondary range symmetric.

Theorem 0.2

Let $A \in ℝ^{n \times n}$ . 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) $N (A^{*}) = N (AV)$
(5) $C (A) = C (A^{S})$
(6) $A^{S} = BA = AC$ where $B$ and $C$ are some nonsingular matrices
(7) $C (A^{*}) = C (VA)$
(8) $C (A^{*}) \oplus N (A) = ℂ^{n}$
(9) $C (A) \oplus N (A^{*}) = ℂ^{n}$

Proof 1

\begin{array}{l} A is secondary range symmetric \Leftrightarrow N (A) = N (A^{S}) \\ \Leftrightarrow N (VA) = N (A^{S} V) (since V^{2} = I) \\ \Leftrightarrow VA is EP \\ \Leftrightarrow V (VA) V^{S} is EP \\ \Leftrightarrow AV is EP . \end{array}

Hence $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ holds true.

$(1) \Leftrightarrow (4)$

\begin{array}{l} A is secondary range symmetric \Leftrightarrow N (A) = N (A^{S}) \\ \Leftrightarrow N (A) = N ({VA}^{*} V) (by definition of secondary transpose) \\ \Leftrightarrow N (A) = N (A^{*} V) \\ \Leftrightarrow A^{*} V = A^{*} {VA}^{(1)} A \\ \Leftrightarrow A^{*} = A^{*} {VA}^{(1)} AV \\ \Leftrightarrow A^{*} = A^{*} {(AV)}^{(1)} AV \\ \Leftrightarrow N (A^{*}) = N (AV) \end{array}

This proves the equivalence of $(1)$ and $(4)$ .

$(3) \Leftrightarrow (5)$

\begin{array}{l} AV is range symmetric \Leftrightarrow C (AV) = C {(AV)}^{*} \\ \Leftrightarrow C (A) = C ({VA}^{*}) (Since V^{*} = V) \\ \Leftrightarrow C (A) = C ({VA}^{*} V) \\ \Leftrightarrow C (A) = C (A^{S}) \end{array}

Hence $(3)$ and $(5)$ are equivalent.

$(2) \Leftrightarrow (6)$

\begin{array}{l} VA is range symmetric \Leftrightarrow {(VA)}^{*} = VAP for some nonsingular n \times n matrix P \\ \Leftrightarrow A^{*} V = VAP \\ \Leftrightarrow {VA}^{*} V = AP \\ \Leftrightarrow A^{S} = AP \\ \Leftrightarrow A = {(AP)}^{S} = P^{S} A^{S} (By property of secondary transpose) \\ \Leftrightarrow A^{S} = {(P^{S})}^{- 1} A \\ \Leftrightarrow A^{S} = KA where K = {(P^{S})}^{- 1} . (Using {(A^{†_{S}})}^{†_{S}} = A) \end{array}

Hence $(2) \Leftrightarrow (6)$ .

$(5) \Leftrightarrow (7)$

\begin{array}{l} C (A) = C (A^{S}) \Leftrightarrow C (A) = C ({VA}^{S} V) \\ \Leftrightarrow C (A) = C ({VA}^{S}) \\ \Leftrightarrow {VA}^{*} = {AA}^{(1)} {VA}^{*} (By [14]) \\ \Leftrightarrow A^{*} = {VAA}^{(1)} {VA}^{*} \\ \Leftrightarrow A^{*} = (VA) {(VA)}^{(1)} A^{*} \\ \Leftrightarrow C (A^{*}) = C (VA) \end{array}

Thus equivalence of $(5)$ and $(7)$ are proved.

$(2)$ $\Leftrightarrow$ $(8)$ .

\begin{array}{l} VA is range symmetric \Leftrightarrow ℂ^{n} = C (VA) \oplus N (GA) \\ = C ({(VA)}^{*}) \oplus N (A) \\ = C (A^{*} V) \oplus N (A) \\ = C (A^{*}) \oplus N (A) \end{array}

Thus equivalence of $(2)$ and $(8)$ are proved.

$(3) \Leftrightarrow (9)$

\begin{array}{l} AV is range symmetric \Leftrightarrow ℂ^{n} = C (AV) \oplus N (AV) \\ = ℂ^{n} = C (AV) \oplus N ({(AV)}^{*}) \\ = C (A) \oplus N (A^{*}) \end{array}

Thus the equivalence of $(3)$ and $(9)$ is proved.

From the following example, it is clear that EP matrices in Minkowski space defined by Meenakshi³ and secondary range symmetric matrices are two different concepts.

Consider a matrix $A = (\begin{array}{c} - 1 & 1 \\ 1 & - 1 \end{array})$ where $A^{S} = (\begin{array}{c} - 1 & 1 \\ 1 & - 1 \end{array})$ . Note that, here the secondary transpose A^S of the matrix A coincides with A. Clearly A is secondary range symmetric (i.e., $N (A) = N (A^{S}) .$ Here, $A$ is secondary normal as well as $ρ (A) = ρ ({AA}^{S})$ . However, the matrix is not range symmetric in Minkowski’s space since $A^{+} = {GA}^{*} G = (\begin{array}{c} - 1 & - 1 \\ - 1 & - 1 \end{array})$ . It is clear that $N (A) \neq N (A^{+})$ .

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 $B = (\begin{array}{c} 1 & - 1 \\ 1 & - 1 \end{array})$ and $B^{S} = (\begin{array}{c} - 1 & - 1 \\ 1 & 1 \end{array})$ . Clearly $B$ is not secondary range symmetric.

Observe that $B$ is range symmetric in Minkowski space. Since, $B^{+} = {GB}^{*} G = (\begin{array}{c} 1 & - 1 \\ 1 & - 1 \end{array})$ so that $N (B) = N (B^{+})$ .

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 $A^{S} = {VA}^{*} V = {VGA}^{+} GV$ .

A necessary condition for a matrix to be a S - EP (secondary range symmetric) is proved here.

Theorem 0.3

Let $A \in ℝ^{n \times n}$ . If $A$ is secondary normal and $ρ (A) = ρ ({AA}^{S})$ , then $A$ is secondary range symmetric.

Proof 2

Since $A$ is secondary normal, ${AA}^{S} = A^{S} A$ . Hence $ρ (A) = ρ ({AA}^{S}) = ρ (A^{S} A) = ρ (A^{S})$ which implies $N (A) = N ({AA}^{S}) = N (A^{S} A) = N (A^{S})$ . Thus $A$ is secondary range symmetric.

A relation connecting range symmetric and secondary range symmetric matrices is given below:

Theorem 0.4

Let $A \in ℝ^{n \times n}$ . Then any two of the following conditions imply the third one.

(1) $A$ is EP.
(2) $A$ is secondary EP.
(3) $C (A) = C (VA)$ .

Proof 3

$(1), (2) \Rightarrow (3)$

Since $A$ is EP, $C (A) = C (A^{*})$ . By condition (7) of Theorem 0.2, $C (A^{*}) = C (VA)$ as $A$ is secondary range symmetric. Hence $A$ is secondary EP $\Leftrightarrow$ $C (A) = C (VA)$ .

(1), (3) ⇒ (2)

Since A is EP, $C (A) = C (A^{*}) .$ Also, from (3) we have $C (A) = C (VA),$ which gives $C (A^{*}) = C (VA) .$ Hence, by Theorem 0.2, A is secondary range symmetric.

$(2), (3) \Rightarrow (1)$

Since $A$ is S - EP, by condition (2) of Theorem 0.2, $VA$ is range symmetric. Hence $C (VA) = C ({(VA)}^{*}) = C (A^{*} V) = C (A^{*})$ . Also, By (3), $C (A) = C (VA)$ from which it follows that $C (A) = C (A^{*})$ . 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 $M = \frac{1}{2} (A + A^{S})$ and $N = \frac{1}{2 i} (A - A^{S})$ . 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 $A \in ℝ^{n \times n}$ , $A$ is secondary range symmetric if and only if $N (A) \subseteq N (M)$ where $M$ is the S - symmetric part of $A$ .

Proof 4

If $A$ is secondary range symmetric, then $N (A) = N (A^{S})$ . For $x \in N (A)$ , $Ax = 0$ and $A^{S} x = 0$ . Hence $Mx = 0$ . Thus, $N (A) \subseteq N (M)$ , then $Ax = 0 \Rightarrow Mx = 0$ and hence $Nx = 0$ . Therefore $N (A) \subseteq N (N)$ . Thus $N (A) \subseteq N (M) \cap N (N)$ . Since, both $M$ and $N$ are $S$ -symmetric, they are secondary range symmetric.

N (M) = N (M^{S}) = N ({VM}^{*} V) = N (M^{*} V)

and

N (N) = N (N^{S}) = N ({VN}^{*} V) = N (N^{*} V)

Now, $N (A) \subseteq N (M) \cap N (N) = N (M^{*} V) \cap N (N^{*} V) \subseteq N [(M^{*} + {iN}^{*}) V] = N (A^{*} V)$ and $ρ (A) = ρ (A^{*}) = ρ (A^{*} V)$ . Therefore, $N (A) = N (A^{*} V) = N ({VA}^{*} V) = N (A^{S})$ . 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 $m \times n$ matrix $A$ , if $A^{†_{S}}$ exists, then $C (A^{†_{S}}) = C (A^{S})$ .

Proof 5

If $A^{†_{S}}$ exists, then $A^{†_{S}} A = {(A^{†_{S}} A)}^{S} = A^{S} {({AA}^{†_{S}})}^{S} {(A^{†_{S}})}^{S}$ and

A^{†_{S}} = A^{†_{S}} {AA}^{†_{S}} = A^{S} {(A^{†_{S}})}^{S} A^{†_{S}} \Rightarrow C (A^{†_{S}}) \subseteq C (A^{S}) .

Further, $ρ (A^{†_{S}}) = ρ (A) = ρ (A^{S})$ . Thus, $C (A^{†_{S}}) = C (A^{S})$ .

Lemma 2

For an $m \times n$ matrix $A$ , if $A^{†_{S}}$ exists, then $C ({AA}^{†_{S}})$ is the projection on $C (A^{†_{S}})$ and $A^{†_{S}} A$ is the projection on $C (A^{†_{S}})$ .

Proof 6

$x \in C (A)$ if and only if $x = Ay = {AA}^{†_{S}} Ay = {AA}^{†_{S}} x$ . By definition 2, ${AA}^{†_{S}}$ being S - symmetric, idempotent is the projection on $C (A)$ . Similarly, $x \in C (A^{†_{S}})$ if and only if $x = A^{†_{S}} {AA}^{†_{S}} y = A^{†_{S}} Ax$ and $A^{†_{S}} A$ is S - symmetric and idempotent. Hence $A^{†_{S}} A$ is the projection on $C (A^{†_{S}})$ .

Theorem 0.6

For an $n \times n$ matrix $A$ , the following are equivalent:

(1) $A$ is secondary range symmetric and $ρ (A) = ρ (A^{2}) .$
(2) $A^{†_{S}}$ exists and $A^{†_{S}}$ is secondary range symmetric.
(3) There exists a symmetric idempotent matrix E such that $AE = EA$ and $C (A) = C (E) .$

Proof 7

$(1) \Rightarrow (2)$ . Since $ρ (A) = ρ (A^{2})$ and $A$ is secondary symmetric, by using Theorem 0.2 we have, $ρ (A^{S} A) = ρ ({BA}^{2}) = ρ (A^{2}) = ρ (A)$ and $ρ ({AA}^{S}) = ρ (A^{2} C) = ρ (A^{2}) = ρ (A)$ . Thus $ρ (A) = ρ ({AA}^{S}) = ρ (A^{S} A)$ . Hence by Thoerem 0.1, it follows that $A^{†_{S}}$ exists, By Lemma 1 and Theorem 0.2, $C (A^{†_{S}}) = C (A^{S}) = C (A) = C ({(A^{S})}^{S}) = C ({(A^{†_{S}})}^{S})$ . Hence $A^{†_{S}}$ is range symmetric. Thus (2) holds.

$(2) \Rightarrow (3)$ . Since $A^{†_{S}}$ exists, by Lemma 1, $C (A^{†_{S}}) = C (A^{S})$ , by equivalence of condition (1) and (5) of Theorem 0.2, $A^{†_{S}}$ is secondary range symmetric which implies that $C (A^{†_{S}}) = C ({(A^{†_{S}})}^{S})$ . Hence $C (A^{S}) = C ({(A^{†_{S}})}^{S})$ . By Lemma 2, it follows that $A^{S} {(A^{S})}^{†_{S}} = {(A^{†_{S}})}^{S} A^{S}$ , hence ${(A^{†_{S}} A)}^{S} = {({AA}^{†_{S}})}^{S}$ . By definition 2, $A^{†_{S}} A = {AA}^{†_{S}} = E$ , is $S$ -symmetric, idempotent and $AE = EA = A$ ; hence $C (A) \subseteq C (E)$ and $ρ (E) = ρ ({AA}^{†_{S}}) = ρ (A)$ , which implies $C (A) = C (E)$ . Thus $(3)$ holds.

$(3) \Rightarrow (1)$ . Since, $E$ is S - symmetric and idempotent, $E^{S} = E = E^{2}$ , by lemma 1.1, $E^{†_{S}}$ exists and $E^{†_{S}} = E$ implies $E$ is the projection on $C (A)$ . For all reflexive g-inverses $A^{r}$ of $A$ , ${AA}^{r} = {EE}^{†_{S}} = E$ . Since $E$ is S - symmetric and idempotent, ${AA}^{r}$ is $S$ -symmetric. Hence by definition 7, $A^{n}$ exists and ${AA}^{n} = {EE}^{†_{S}} = E$ which implies $EA = A$ . By hypothesis $AE = EA = A$ . Therefore ${AA}^{n} = A^{n} A = E$ . Thus both ${AA}^{n}$ and $A^{n} A$ are S - symmetric. By definition 2, $A^{†_{S}}$ exists and $E = {AA}^{†_{S}} = A^{†_{S}} A$ . By taking secondary transpose on $AE = EA = A$ , we get ${EA}^{S} = A^{S} E = A^{S}$ . $C (A^{S}) \subseteq C (E) = C (A)$ and $ρ (A^{S}) = ρ (A)$ . Therefore $C (A) = C (A^{S})$ . By theorem 0.2, $A$ is secondary ramge symmetric. $ρ ({AA}^{S}) \geq ρ ({AA}^{S} {(A^{†_{S}})}^{S}) = ρ (A {(A^{†_{S}})}^{S})$ , $ρ (AE) = ρ (A) \geq ρ ({AA}^{S})$ . Thus $ρ (A) = ρ ({AA}^{S}) = ρ (A^{2} C) = ρ (A^{2})$ . Thus (1) holds. Hence the theorem.

Corollary 1

Let A be $n \times n$ secondary range symmetric matrix. Then exists $A^{†_{S}}$ if and only if $ρ (A) = ρ (A^{2})$ .

Proof 8

Since $A$ is secondary range symmetric and $ρ (A) = ρ (A^{2})$ , 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 $ρ (A) = ρ ({AA}^{S}) = ρ (A^{S} A)$ and by Theorem 0.2, $A^{S} = AC$ . Hence $ρ (A) = ρ (AAC) = ρ (A^{2})$ .

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.

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18. Xing Z: On the deterministic and nondeterministic Mueller matrix. J. Mod. Opt. 1992; 39:461–484. Publisher Full Text
19. Renardy M: Singular value decomposition in Minkowski Space. Linear Algebra Appl. 1996; 236:53–58. Publisher Full Text
20. Purushothama DS: Sum of secondary range symmetric matrices. Glob. Stoch. Anal. 2024; 11(3).

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Version 2

VERSION 2 PUBLISHED 19 Feb 2024

Author details Author details

Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, Karnataka, 576104, India

Divya Shenoy
Roles: Conceptualization, Data Curation, Formal Analysis, Validation, Writing – Original Draft Preparation

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (2)

version 2

Revised

Published: 11 Sep 2024, 13:112

https://doi.org/10.12688/f1000research.144171.2

version 1

Published: 19 Feb 2024, 13:112

https://doi.org/10.12688/f1000research.144171.1

© 2024 Shenoy D. This is an open access article 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.

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Shenoy D. Secondary range symmetric matrices [version 2; peer review: 3 approved]. F1000Research 2024, 13:112 (https://doi.org/10.12688/f1000research.144171.2)

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ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions

Version 2

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PUBLISHED 11 Sep 2024

Revised

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Reviewer Report 11 Oct 2024

Pankaj Kumar Manjhi, Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India

Approved

https://doi.org/10.5256/f1000research.168149.r322688

The author has successfully addressed all the provided suggestions, resulting in a significantly improved manuscript. The revised version is now clearer, more comprehensible, and effectively presents the research. It is suitable for proceeding to the next stage of the publication ... Continue reading

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Version 1

VERSION 1

PUBLISHED 19 Feb 2024

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Reviewer Report 08 Jun 2024

Mehsin Jabel Atteya, Department of Mathematics, Al- Mustansiriyah University, Falastin St, Baghdad, Iraq

Approved

https://doi.org/10.5256/f1000research.157925.r261048

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 ... Continue reading

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
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: 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.

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Report a concern

Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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.
... Continue reading 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.
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.
Competing Interests: There are no competing interests to declare Close
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COMMENTS ON THIS REPORT

Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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.
... Continue reading 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.
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.
Competing Interests: There are no competing interests to declare Close
Report a concern

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Reviewer Report 08 Jun 2024

Pankaj Kumar Manjhi, Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India

Approved with Reservations

https://doi.org/10.5256/f1000research.157925.r274406

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

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

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

Partly
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
Are the conclusions drawn adequately supported by the results?

Partly

Competing 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.

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Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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 ... Continue reading 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.
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.
Competing Interests: There are no competing interests to declare Close
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Respond or Comment

COMMENTS ON THIS REPORT

Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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 ... Continue reading 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.
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.
Competing Interests: There are no competing interests to declare Close
Report a concern

Views

Reviewer Report 28 Mar 2024

P Sam Johnson, National Institute of Technology Karnataka, Surathkal, Surathkal, Karnataka,, India

Approved

https://doi.org/10.5256/f1000research.157925.r252908

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
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

References

1. Johnson P, Vinoth A: Product and factorization of hypo-EP operators. Special Matrices. 2018; 6 (1): 376-382 Publisher Full Text
2. Johnson P: Closed EP and hypo-EP operators on Hilbert spaces. The Journal of Analysis. 2022; 30 (4): 1377-1390 Publisher Full Text

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.

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Report a concern

Author Response 04 Apr 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

04 Apr 2024

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests. Close
Report a concern
Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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 ... Continue reading 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.
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: There are no competing interest to declare. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 04 Apr 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

04 Apr 2024

Author Response

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 ... Continue reading 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.
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.
Competing Interests: No competing interests. Close
Report a concern
Author Response 11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

11 Sep 2024

Author Response

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 ... Continue reading 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.
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: There are no competing interest to declare. Close
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VERSION 2 PUBLISHED 19 Feb 2024

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P Sam Johnson, National Institute of Technology Karnataka, Surathkal, Surathkal, India
Pankaj Kumar Manjhi, Vinoba Bhave University,, Hazaribag,, India
Mehsin Jabel Atteya, Al- Mustansiriyah University, Falastin St, Iraq

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Reviewer Report

1 Views

11 Oct 2024 | for Version 2

Pankaj Kumar Manjhi, Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India

1 Views Cite this report Responses(0)

Approved

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No competing interests were disclosed.

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.

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13 Views

08 Jun 2024 | for Version 1

Mehsin Jabel Atteya, Department of Mathematics, Al- Mustansiriyah University, Falastin St, Baghdad, Iraq

13 Views Cite this report Responses(1)

Approved

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
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

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.

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Author Response

11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

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.

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Reviewer Report

12 Views

08 Jun 2024 | for Version 1

Pankaj Kumar Manjhi, Department of Mathematics,, Vinoba Bhave University,, Hazaribag,, Jharkhand,, India

12 Views Cite this report Responses(1)

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

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

Partly
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
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Combinatorial matrices, Discrete Mathematics, computer science

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Author Response

11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

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.

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Reviewer Report

27 Views

28 Mar 2024 | for Version 1

P Sam Johnson, National Institute of Technology Karnataka, Surathkal, Surathkal, Karnataka,, India

27 Views Cite this report Responses(2)

Approved

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

References

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.

Respond to this report

Responses (2)

Author Response

11 Sep 2024

Divya Shenoy, Department of Mathematics, Manipal Institute of Technology, Manipal, Manipal Academy of Higher Education, Udupi, 576104, India

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.

View more View less

Competing Interests

There are no competing interest to declare.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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[12] 12. Savitha V, Shenoy DP, Umashankar K, et al.: Secondary transpose of a matrix and generalized inverses. J. Algebra Its Appl. 2024; 23(3): 2450052. Publisher Full Text

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Secondary range symmetric matrices

Abstract

Keywords

Revised Amendments from Version 1

Introduction

Preliminaries

Definition 1

Definition 2

Definition 3

Definition 4

Theorem 0.1

Definition 5

Definition 6

Results

Definition 7

Definition 8

Definition 9

Theorem 0.2

Proof 1

Theorem 0.3

Proof 2

Theorem 0.4

Proof 3

Theorem 0.5

Proof 4

Lemma 1

Proof 5

Lemma 2

Proof 6

Theorem 0.6

Proof 7

Corollary 1

Proof 8

Conclusion

Data availability

References

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