Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# The Spectrum of the Orthogonal Matrix with its Cube being Symmetric

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. 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. Preprints 2023, 2023050434. 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

Comment 1
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Comment: 1 The title is “The spectrum of the orthogonal matrix with its cube be symmetrc”.Here “be” has been replaced by “being”.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”.
+ Respond to this comment