The Spectrum of the Orthogonal Matrix with its Cube be Symmetric

Mei Xiang Chen; Zhong Peng Yang; Zhi Xing Lin

doi:10.20944/preprints202305.0434.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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)

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.

orthogonal matrix; cube; symmetric; trace; sum

Computer Science and Mathematics, Applied Mathematics

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The Spectrum of the Orthogonal Matrix with its Cube be Symmetric

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