The \(\mathcal {L_C}\)-Structure-Preserving Algorithms of Quaternion \(LDL^H\) Decomposition and Cholesky Decomposition

Abstract

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.