3 Quaternion Representation 5 3.1 Axis-Angle to Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Quaternion to Axis-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Quaternion to Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Matrix to Quaternion . . . . . . . . . . ....

A necessary and sufficient condition is given for the quaternion matrix equations AiX + Y Bi = Ci (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

The main purpose of this paper is to show the application of mathematical system of quaternions in physics. We show a basic way of generating a quaternion operator and a method of obtaining the Schrodinger’s equation using this operator matrix. Then we investigate the various probable uses of the coefficient matrix to scale relativity and spacetime quantization.

A necessary and sufficient condition is given for the quaternion matrix equations AiX + Y Bi = Ci (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

Attitude quaternion [1] is the attitude parameterization of choice for spacecraft attitude estimation for several reasons: 1) it is free of singularities, 2) the attitude matrix is quadratic in the quaternion components, and 3) the kinematics equations is bilinear and an analytic solution exists for the propagation. However, the components of the attitude quaternion are not independent of each ...

We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to given generalized reflection matrix. adopt real representation method derive solutions it is solvable. Moreover, we obtain explicit expressions of least-squares solutions.

The main aim of this paper is to study quaternion matrix factorization for low-rank completion and its applications in color image processing. For the real-world images, we proposed a novel model called (LRQC), which adds total variation Tikhonov regularization factor matrices preserve spatial/temporal smoothness. Moreover, proximal alternating minimization (PAM) algorithm was tackle correspond...