New Algebra Structure And Applications Of Quaternionic Matrix

In 1843, Hamilton who was a British mathematician put forward the concept of quaternion first. In recent 30 years, many experts and scholars were carrying on an extensive research about quaternionic matrix and got plenteous theoretical results. But the research work of quaternion was largely restricted due to the non-commutation of quaternionic multiplication, the numerical algorithm of quaternionic matrix was really rare. Recent computing software, include MATLAB which can deal with matrix operation very well, can't find relative software package about the operation of quaternionic matrix. Therefore, studying the new representation of quaternionic matrix and seeking an algebra theory which is suitable for recent computing software have much practical value.In this paper, we elaborate the definition and basic properties of quaternion first. In the meantime, we introduce the real representation and complex representation of quaternion. Then we introduce the definition and relative properties of the real representation and complex representation of quaternionic matrix. Later, a concept of new companion vector is introduced and its properties are discussed also. Then we study the relation between new companion vector and companion vector. We also discuss the numerical features of quaternionic matrix, which include right characteristic value and rank. We introduce the definition and relative properties of the generalized inverse of quaternionic matrix also. By using structure preserving property of the real representation and complex representation of the quaternionic matrix, we give numerical algorithm of calculating Moore-Penrose inverse of the quaternionic matrix and solving a type of the quaternionic matrix equation AXB = C. We give the numerical algorithm of calculating characteristic decomposition of quaternionic matrix also.