Structure-preserving QR Algorithm Of General Quaternion Eigenvalue Problem With Application To Color Watermarking

New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the(right)eigenproblem of general quaternion matrices.Under the orthogonally J RS-symplectic transformations,the J RS-QR algorithm are firstly proposed for J RS-symmetric matrices and then applied to calculate the Schur forms of quaternion matrices.A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices.For the eigenvalue problem of quaternion matrices,we propose a new structure-preserving eigen solver.The solver efficiently solves the quaternion eigenvalues and their structural eigenvectors under the J RS-symmetry structure.Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed eigen solver.Finally,we propose an efficient and stable quaternion structure-preserving SVD algorithm,and apply the quaternion Schur decomposition and singular value decomposition to the color image watermarking technique,using quaternions to represent the three primary colors.The quaternion is used to represent the integrity of the three primaries.The three color channel links are strengthened,and the quality of the color watermark is also enhanced.