ℒ_𝒞 structure-preserving method based on semi-tensor product of matrices for the QR decomposition in quaternionic quantum theory

In this paper, based on semi-tensor product of matrices, a novel structure-preserving algorithm called ℒ_𝒞 structure-preserving algorithm is applied to quaternion Householder transformations for the first time. Then by means of the above contents, we present the ℒ_𝒞 structure-preserving methods for the QR decomposition and column pivoting QR decomposition of quaternion matrices, which make full use of high-level operations and relation of operations between quaternion matrices and their ℒ_𝒞 representation matrices. The numerical examples show that compared with the ‘qr’ of the Quaternion toolbox, the ℒ_𝒞 structure-preserving algorithm is more effective in calculating the QR decomposition of quaternion matrices.