Distribution of Eigenvalues and Upper Bounds of the Spread of Interval Matrices

Response to Reviewer 1 Comments

1. Summary

Thank you very much for taking the time to review this manuscript and affording us an opportunity to revise our manuscript for publication. All the comments are very valuable and helpful for revising and improving our paper, as well as the important guiding significance to our further researches.

Please find the detailed responses below and the corresponding revisions.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Can be improved

We enhanced the abstract by providing more context on the significance of the distribution of eigenvalues and the upper bounds of interval matrices, along with a brief mention of the key methodologies used in our study.

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Can be improved

Are the methods adequately described?

Can be improved

We revised the abstract section to provide a more detailed and comprehensive description of the methodologies employed in our research.

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Must be improved

We provided a section entitled Conclusion to ensure that they are more closely aligned with the results presented in the manuscript.

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1:   **Abstract:** The abstract is concise but could be improved by providing a bit more context regarding why the distribution of eigenvalues and the upper bounds of interval matrices are significant. Additionally, it would be helpful to briefly mention the key methodology or techniques used in the paper.

Response 1: We enhanced the abstract by providing more context on the significance of the distribution of eigenvalues and the upper bounds of interval matrices, along with a brief mention of the key methodologies used in our study.

Comments 2: **Language Editing:** The paper requires some language editing to enhance readability and clarity.

Response 2: We appreciate your language editing suggestions. We have made the recommended revisions for improved readability and clarity.

Comments 3: **Organization and Flow:** Ensure that the paper flows logically from one section to the next. Each section should build upon the previous one and lead the reader through the research process. Consider using clear transitional sentences or phrases to guide the reader.

Response 3: We have enhanced the logical progression between sections and provide clear transitional sentences to guide the reader.

Comments 4: Provide a section entitled Conclusion.

Response 4: We appreciate your suggestion to include a Conclusion section. We will add a dedicated Conclusion section to the manuscript.

Comments 1: The manuscript has many and significant problems. The language is very poor and the presentation of the work rather amateur. It contains a lot of typos and many sentences are difficult to understand their meaning. For example

"For fixed λ , aii(i = 1, 2, · · · , n) locates on x axis, so we can get numerous small discs that with centered..."

"Because the spread of a matrix describes the maximum of distance of any two eigenvalues." A sentence stop without any explanation.

"whose elements belong to a fixed interval [−a, a] are established, it has been shown in [11, Theorem 7] that if A ∈ Sn[−a, a] with n ⩾ 2 and a > 0, then..."

Response 1: We appreciate your language editing suggestions. We have made the recommended revisions for improved readability and clarity.

Comments 2: The references are very old. For example, in the manuscript is written that "Recently, some researchers started to set foot on the study of eigenvalues of interval matrices (see Ref [10]-[16])" and all the references are more than 30 years back except one that is 15 years old.

Response 2:  We removed Theorem 2.1 and modified Theorem 3.1 to Theorem 3.3 carefully . And we also provide two other examples to  demonstrate the correctness of the results. 

Comments 3: Finally, in the only example that was given to numerically verify the Theorem 3.3 it has typos and inaccuracies. For example r_i is probably R_i and  min p_ii is not -3. Moreover  if r_i=R_i why the maximum is 3? In addition, and most important,  if I translate and calculate  correctly the formula of Theorem 3.3, gives s(a)<11.53 and not 5.64. Thus, the bound is not better than the already know.

Response 3:  We updated the references and provided the latest references.

Comments 4: The references are very old. For example, in the manuscript is written that "Recently, some researchers started to set foot on the study of eigenvalues of interval matrices (see Ref [10]-[16])" and all the references are more than 30 years back except one that is 15 years old.

Response 4:  We revised Theorem 3.2 and Theorem 3.3 and provided corresponding examples to illustrate my results.

Comments 5: The quality of the English and the overall presentation of the manuscript need serious revision.

Response 5:  We have made the revisions for improved readability and clarity.

Response to Reviewer 2 Comments

1. Summary

Thank you very much for taking the time to review this manuscript and affording us an opportunity to revise our manuscript for publication. All the comments are very valuable and helpful for revising and improving our paper, as well as the important guiding significance to our further researches.

Please find the detailed responses below and the corresponding revisions.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Not Applicable

Are the methods adequately described?

Yes

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Yes

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1:  This paper concerns the distribution of eigenvalues of interval matrices, that is, matrices where each entry is within some interval - this interval may differ for each entry. It also concerns the spread of interval matrices, that is, the largest difference (after taking the modulus) between any two eigenvalues.

The results within the paper are interesting and the paper should definitely be considered for publication. However, the English language used within it is quite subpar, which detracts from the overall presentation and understanding of the paper. It is recommended that the paper is forwarded to an appropriate grammar checker before it is published.

Response 1: We would like to express our gratitude for your constructive review and positive feedback on the content of our paper. We sincerely appreciate your acknowledgment of the significance of our research.

We take your concerns about the quality of English language usage seriously, and we are committed to improving the clarity and readability of our manuscript. We agree that proper language editing is essential to enhance the overall presentation and understanding of the paper. We ensure that the manuscript undergoes thorough language editing before final submission, with the aim of meeting the highest language standards for publication.

Your valuable feedback is greatly appreciated, and it will undoubtedly contribute to the refinement of our paper.

4. Response to Comments on the Quality of English Language

Point 1: The English language used in the paper should be improved. Here are a few of the grammar mistakes encountered:

1) "has been widely paid attention to" -> "is a widely-researched topic"

2) "he obtained some beneficial equalities" - I don't know what the authors mean by this.

3) "only a few people concerned the spread" -> "only a few of them were interested in the spread"

4) "and Corollary 2.1 has been obtained by Theorem 2.2." - I don't understand what is meant by this.

5) "must be lied in" -> "must lie in"

and so on.

Response 1:  We appreciate your feedback regarding the quality of English language usage in our paper. We agree that addressing these grammar issues is essential to enhance the clarity and readability of the manuscript. We have made the necessary revisions to correct these language errors and improve the overall quality of the writing.