Version 1
: Received: 4 May 2023 / Approved: 8 May 2023 / Online: 8 May 2023 (03:42:35 CEST)
Version 2
: Received: 26 May 2023 / Approved: 26 May 2023 / Online: 26 May 2023 (15:59:50 CEST)
How to cite:
Chen, M.; Yang, Z.; Lin, Z. The Spectrum of the Orthogonal Matrix with its Cube being Symmetric. Preprints2023, 2023050434. https://doi.org/10.20944/preprints202305.0434.v2
Chen, M.; Yang, Z.; Lin, Z. The Spectrum of the Orthogonal Matrix with its Cube being Symmetric. Preprints 2023, 2023050434. https://doi.org/10.20944/preprints202305.0434.v2
Chen, M.; Yang, Z.; Lin, Z. The Spectrum of the Orthogonal Matrix with its Cube being Symmetric. Preprints2023, 2023050434. https://doi.org/10.20944/preprints202305.0434.v2
APA Style
Chen, M., Yang, Z., & Lin, Z. (2023). The Spectrum of the Orthogonal Matrix with its Cube being Symmetric. Preprints. https://doi.org/10.20944/preprints202305.0434.v2
Chicago/Turabian Style
Chen, M., Zhongpeng Yang and Zhixing Lin. 2023 "The Spectrum of the Orthogonal Matrix with its Cube being Symmetric" Preprints. https://doi.org/10.20944/preprints202305.0434.v2
Abstract
The orthogonal matrix with cube be symmetric is a common class of matrices with important properties. We pay attention to this kind of matrices. By using the close relationship between the eigenvalues of a matrix and the trace of its power, we obtain the algorithm for its all possible different eigenvalues and multiplicities. The calculation formula is expressed only by the trace of a matrix and its power, avoiding solving the characteristic polynomial. The method is simple and practical. Furthermore, a new essential characterization for the sum of orthogonal matrix pairs being orthogonal is given as well.
Keywords
orthogonal matrix; cube; symmetric; trace; sum
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received:
26 May 2023
Commenter:
Meixiang Chen
Commenter's Conflict of Interests:
Author
Comment:
1 The title is “The spectrum of the orthogonal matrix with its cube be symmetrc”.Here “be” has been replaced by “being”. 2 In Page 1, Line 3,“Computer software such as Mathematica and Maple”,“software” should be “softwares”. 3 In Page 1, Line 12, “Smith O.K. have obtained...”, “have” should be “has”.
Commenter: Meixiang Chen
Commenter's Conflict of Interests: Author
2 In Page 1, Line 3,“Computer software such as Mathematica and Maple”,“software” should be “softwares”.
3 In Page 1, Line 12, “Smith O.K. have obtained...”, “have” should be “has”.