Matrix Decompositions Over Quaternion Field

We studys the theory of matrix decomposition and matrix equation over quaternion field, along the clues of the theory of matrix decomposition and matrix equation over real number field and complex number field. It gets some results and enrichs quaternion matrices theory.The main results of this paper lie in:1. By introducing a new LU decomposition of quaternion matrix, we establish the relationship between this new LU decomposition of quaternion matrix and the corresponding part of complex matrix. Consequently, we obtain this new LU decomposition existence determinant condition and obtain the method of calculation.2. We give the necessary and sufficient condition of quaternion matrices to have unique LDU decompositions and build a kind of practical algorithm of matrix LU decompositions, over including Crout decompositions and Doolittle decompositions.3. We extend some properties and conclusions of complex number matrix equation AXB = C over quaternion field and give the necessary and sufficient condition of which the quaternion matrix equation AXB = C can be solved. Moreover, we describe the structure of solutions of such an equation under circumstances that one of A and B is real matrix, other is self-conjugate matrix.