The Research Of Matrix Decomposition And Partial Order

The study of matrix partial order and matrix decomposition is a very important part of matrix theory system.Partial order problem not only has an important position in theory,but also plays a great role in promoting the development of other applications and disciplines,such as the study of observable measurements and states in quantum physics,the application of control systems or linear models.Based on a large number of previous research achievements and foundation,this thesis further enriched the theoretical system of matrix partial order by defining a new matrix decomposition and describing a new matrix partial order.This thesis will be introduced from the following aspects:Firstly,a new theorem of matrix decomposition,core-EP-nilpotent decomposition,is presented,and its related properties are investigated.Further,based on the new matrix decomposition,two new matrix partial orders are described: core-E-N partial order and core-ES partial order.Meanwhile,the relevant properties of the two new partial orders are studied and the relationship between the two new partial orders and minus partial order is illustrated with examples.Secondly,a new matrix partial order is described: CS partial order.Through the existing polar decomposition,this partial order is characterized by the combination of core partial order and star partial order,and the properties of CS partial order and the relationship between CS partial order and some existing partial orders are studied with examples.Finally,on the basis of describing the specific form of CS partial order,a more general partial order,WC partial order,is described by using the existing polar-like decomposition.The nature of WC partial order is studied and the relationship between WC partial order and some existing partial orders is illustrated with examples.The relationship between WC partial order and CS partial order is explained by notes.Finally,some potential applications of CS partial order and WC partial order in quantum physics are discussed.