Decomposition Of Quaternion Matrix And Its Application

Since English mathematician W.R. Hamilton discovered quaternion, the applications of quaternion and quaternion matrix have been very abroad in geostatics, the theory of peg-top using, strap-down inertial, navigation, enginery and organ, robot technology and artificial satellite attitude control, and so on. So it has been pop field at home and abroad in the last twenty years.Decomposition of a matrix is to transform a given matrix into two or three matrices which are of normal form. As known to all, it plays a very important part in matrix theory and computation on real or complex field. Accordingly, it is also very important for quaternion matrix theory.In this paper, decomposition of a quaternion matrix is systemically studied. To one quaternion matrix, according to its normal form, into which a quaternion matrix is transformed after decomposed, the types of decomposition of a quaternion matrix are divided into three large classes: diagonalized decomposition, triangular decomposition and triangular-diagonalized decomposition. Furthermore, we discuss the decomposition of quaternion matrix pencil based on the classes.For the main results, we have:(1) LU decomposition of a quaternion matrix is firstly given out. And then we have LDU decomposition, Cholesky decomposition, quasi-LU decomposition, quasi-LDU decomposition, LDM~T decomposition and LDL~* decomposition. We also give out Schur decomposition for normal quaternion matrix.(2) We work out generalized Schur decomposition of quaternion matrix pencil and give out a sufficient condition for simultaneous unitary diagonalization of two quaternion matrices.