Abstract: Quaternion is set up by W.R.Hamilton, a British mathematician, in1843, and it has become a new mathematical discipline in subsequent studies step by step. As a new phenomenon, quaternion has attracted deeply mathematicians and physicists all over the world. The studies on quaternion and quaternion algebra have never ceased nearly 200 years. In particular, the applications of quaternion and quaternion matrix in rigid body dynamics, quantum physics, system simulation, computer graphics image processing and satellite attitude control and so on, make them more and more prominent in the last 30 years. Quaternion has become gradually an important tool for other subjects from a math phenomenon, and the researches about its theory and applications have been an important aspect of branch of mathematics at present. Therefore, it is necessary to study the quaternion algebra, and this dissertation is carried out on these grounds. In this dissertation, we mainly discuss the trace and norm theory, the rank and eigenvalues of quaternion matrix and obtain some helpful results. For the main contents, we have:(1) We present the lower bound for the rank of quaternion matrix with the help of that on complex matrix. Based on this, a sufficient condition for the singularity of quaternion matrix is obtained.(2) By use of the complex representation of quaternion matrix, the problem of the existence for right eigenvalues of quaternion matrix is changed into eigenvalues of complex matrix, and it is well solved. Moreover, the specific solution is given.(3) We give the estimation of the upper bound for the real and imaginary parts of quaternion matricesâ€™standard (right) eigenvalues. Further, some estimation theorems about the left eigenvalues of quaternion matrix are given by using the generalized Ger?chgorin theorem and the particle and centre of gravity theory.(4) For self-conjugate quaternion matrix, by using the characteristic of eigenvalues and trace, it is shown that the estimation of sums of partial eigenvalues and two necessary and sufficient conditions concerning the inequalities of the trace for self-conjugate quaternion matrix…

Key words: Quaternion Matrix; Rank; Estimation of Eigenvalues; Norm; Trace

# Study on Some Algebra Questions of Quaternion Matrix

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