In this paper, the \(\mathcal {L_C}\)-structure-preserving algorithms of \(LDL^H\) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices based on the semi-tensor product of matrices are studied. We first propose \(\mathcal {L_C}\)-representation by using the semi-tensor product of matries and the structure matrix of the product of the quaternion. Then, \(\mathcal {L_C}\)-structure-preserving algorithms of \(LDL^H\) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices are proposed by using \(\mathcal {L_C}\)-representation, and the advantages of our method are obtained by comparing the operation time and error with the real structure-preserving algorithms in Wei et al. (Quaternion matrix computations. Nova Science Publishers, Hauppauge, 2018). Finally, we apply the \(\mathcal {L_C}\)-structure-preserving algorithm of Cholesky decomposition to strict authentication of color images.

本文研究了基于矩阵半张量积的四元数Hermitian正定矩阵的\(\mathcal {L_C}\)结构保持算法\(LDL^H\)分解和Cholesky分解。我们首先使用矩阵的半张量积和四元数乘积的结构矩阵提出\(\mathcal {L_C}\)表示。然后，利用\(\mathcal {L_C}\)表示，提出了四元数Hermitian正定矩阵的\ (\mathcal {L_C}\ )分解和Cholesky分解的结构保持算法，通过与 Wei 等人的真实结构保持算法的运行时间和误差进行比较，得出我们的方法的优点。 （四元数矩阵计算。Nova Science Publishers，Hauppauge，2018）。最后，我们将Cholesky 分解的\(\mathcal {L_C}\)结构保持算法应用于彩色图像的严格认证。