Some Results on Majorization of Matrices

Round 1

Reviewer 1 Report

This paper describes some results on matrices and their majorization. The authors discussed sufficient conditions for the majorization of matrices and discussed some relevant theorems. The work is more abstract and it can interesting if the author addresses the following points.

1. Please add more lines about the subject matter in the abstract.
2. Please explicitly describe the research gap and combine this research with previous research. 
3. The introduction needs attention as the material on the subject matter is very little. Please add more relevant literature for the reader and use references in sequential order.
4. Please state (if possible ) the possible applications of the work.
5. Please add more examples in the main body of the file to support the theorems. 
6. Is there any limitation of the study? what is it? describe it in the discussion section.
7. what is the future aspects of the paper.

 

 

 

Author Response

Comments and suggestions for authors            

This paper describes some results on matrices and their majorization. The authors discussed sufficient conditions for the majorization of matrices and discussed some relevant theorems.  The work done is more abstract and it can be interesting if the author addresses the following points.

S.No	Reviewer’s Comments/Suggestions	Authors’ Responses
1	Please add more material on the subject matter in the abstract.	More material added in the abstract.
2	Please explicitly describe the research gap and combine this research with previous research.	Research gap is discussed in a detailed manner at the end of section 1.
3	The introduction needs attention as the material on the subject matter is very little. Please add more relevant literature for the reader and use references in sequential order.	Introduction section is modified and few more relevant research papers were added in the paper.
4	Please state (if possible ) the possible applications of the work.	Matrix majorization has several applications in science and engineering section which is described in the introduction section.
5	Please add more examples in the main body of the file to support the theorems.	Example for each theorem is given in the paper.
6	Is there any limitation of the study? what is it? describe it in the discussion section.	We obtained necessary conditions and sufficient conditions for strong and doubly stochastic majorizations. Finding necessary and sufficient conditions will be an open problem. This was described in the Conclusions section.
7	What is the future aspects of the paper	Giving characterizations of other majorizations and finding necessary and sufficient conditions for other majorizations are the future scope.

 

Author Response File: Author Response.docx

Reviewer 2 Report

For two n × m real matrices X and Y , X is said to be majorized by Y , written X ≺ Y
if X = SY for some doubly stochastic matrix of order n. In this paper, the authors
obtained some necessary and sufficient conditions for X, Y to satisfy X ≺ Y for the case
where rank Y = n − 2 and in general for rank Y = n − k where 1 ≤ k ≤ n − 1. Also they
obtained some necessary and sufficient conditions for X to be majorized by Y with some
conditions on X and Y .
The paper is well written and the proofs are correct. I recommend the paper for
publication.
Following are minor typo errors.
1. Page 2, line 35, In [5], ....
2. Page 2, Theorem 2.1 line 5, last term should be (1 − σ(y2))z2.
3. Page 3, line 74, z1, z2, ..., zk.
4. Page 5, Proof of the theorem 2.5 ends at line 180.
5. Page 6, lines 216 and 217, x − − > X.
6. Page 6, line 219 i = 1, 2, ..., m.
7. Page 7, line 258, b = 1 and ....

Author Response

Comments and suggestions for authors

For two n × m real matrices X and Y , X is said to be majorized by Y , written X ≺ Y
if X = SY for some doubly stochastic matrix of order n. In this paper, the authors
obtained some necessary and sufficient conditions for X, Y to satisfy X ≺ Y for the case
where rank Y = n − 2 and in general for rank Y = n − k where 1 ≤ k ≤ n − 1. Also they
obtained some necessary and sufficient conditions for X to be majorized by Y with some
conditions on X and Y .
The paper is well written and the proofs are correct. I recommend the paper for
publication.
Following are minor typo errors.

S.No	Reviewer’s Comments/Suggestions	Authors’ Responses
1	Page 2, line 35, In [5], ....	Corrected
2	Page 2, Theorem 2.1 line 5, last term should be (1 − σ(y2))z2.	Corrected
3	Page 3, line 74, z1, z2, ..., zk.	Corrected
4	Page 5, Proof of the theorem 2.5 ends at line 180.	Corrected
5	Page 6, lines 216 and 217, x − − > X.	Corrected
6	Page 6, line 219 i = 1, 2, ..., m.	Corrected
7	Page 7, line 258, b = 1 and ....	Corrected

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

"Author Contributions: Both the authors have contributed equally to this paper." but in the authors list of the paper double dagger symbol appear only at the first author.

References: 7 references for the subject is definitely not enough. Also the refs should be written using the publisher's style.

The research aim should be better introduced.

Some (if not all) of the results given in "2. Matrix Majorization" already appears in the literature, even if not with the same math formalism. The connection must be given.

Majorization of vectors and matrices plays an important role in order statistics (see for instance A test detecting the outliers for continuous distributions ...) and in eigenvector/eigenvalue problems (see for instance The eigenproblem translated for alignment ...) analysis. The authors should mention such important connections. Also majorization of (0,1) matrices have important applications in classification theory and principal/dominant component analysis.

 

Author Response

Comments and suggestions for authors

S.No	Reviewer’s Comments/Suggestions	Authors’ Responses
1	"Author Contributions: Both the authors have contributed equally to this paper." but in the authors list of the paper double dagger symbol appear only at the first author.	The double dagger symbol is added for both authors.
2	References: 7 references for the subject is definitely not enough. Also the refs should be written using the publisher's style.	Few more references are added related to the paper. References are also written in a Publishers’ style.
3	The research aim should be better introduced.	Aim of this paper is to study necessary and sufficient conditions for matrix majorization. This was stated in abstract as well as in Introduction section.
4	Some (if not all) of the results given in "2. Matrix Majorization" already appears in the literature, even if not with the same math formalism. The connection must be given.	All the theorems stated in section 2 and section 3 are new and they are generalized version of the existing theorems. This is stated clearly in the Introduction section.
5	Majorization of vectors and matrices plays an important role in order statistics (see for instance A test detecting the outliers for continuous distributions ...) and in eigenvector/eigenvalue problems (see for instance The eigenproblem translated for alignment ...) analysis. The authors should mention such important connections. Also majorization of (0,1) matrices have important applications in classification theory and principal/dominant component analysis.	Some applications of matrix Majorization are mentioned in the Introduction section. We thank the reviewer for citing some of the applications of matrix majorization.

 

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I cannot see any changes as the authors did not highlight anything. Can you send me a version where I can see the changes highlighted?

 

Reviewer 3 Report

The authors did not made any substantial changes. I am sorry, but I cannot support publication of the manuscript in the journal.

Round 3

Reviewer 1 Report

The paper can be accepted now.

Reviewer 3 Report

The only change visible is around the research aim. Still, the authors did not made any substantial changes.

The references list is very very short for the selected subject ("matrix majorization"). My search on "matrix majorization" retrieved no less than 1545 results on MDPI ("https://www.mdpi.com/search?q=matrix+majorization") while the authors was not able to find any of these results.

The authors are not able to follow the journal style for the references. It is now clear for me that they never opened any research published by MDPI, but they want to publish with it. How so?

I am sorry, but I cannot support publication of the manuscript in the journal.