On Some Problems Of Quaternion Matrices

The quaternion algebra and quaternion matrix have great applications in the field of mathematics, physics, engineering technology, information science, etc. With the development of science technology and application of computer, the application of quaternion and quaternion matrix has become a hot topic in the field of engineering technology. Because of the non-commutation of the quaternion multiplication, it is difficult to study the quaternion matrices. Moreover, there are still some valuable topics to be studied.In this paper, we study some important problems for the algebra structures of quaternion algebra, the decomposition, the condition of centralizable quaternion ma-trix, the properties and calculations of left or right eigenvalues of quaternion matrix. This paper is composed of three chapters as follows.In Chapter1, we introduce the historical background and present research situ-ation, make a survey of the methods used in the study of quaternion and quaternion matrices and introduce the main research results of this paper.In Chapter2, we give the definition for an algebra with principal axis property: Let F be a field, R be an F-algebra with an involution aâ†'a. If every Hermite matrix A over R is unitary similar to a diagonal matrix over F, then R is called an algebra with the principal axis property. Moreover, we give several equivalent definitions of an algebra with the principal axis property, and show that the algebra with the principal axis property is equivalent to the algebra with the SVD(singular value decomposition) property.so if R+is a Galois order closed field,then (-1,-1/R+) over R+has the principal axis property. We also discuss the theory of unitary similarity of matrices over the algebra with the principal axis property, and some properties of the Galois order closed field.In Chapter3, we give the definition of J-self-conjugate quaternion matrix and extend the decomposition theorem of centralizable quaternion matrix. Then we extend the research results about eigenvalues problems of quaternion matrix to the matrix of the algebra with the principal axis property, and discuss the properties and calculations of left (right) eigenvalues of quaternion matrix over Galois order dosed field.