On The Block Independency Of Two Matrices And The Structural Representation For The Generalized Inverses Of Two Matrices

Generalized inverse matrix is an important subject in matrix analysis.Blocked matrix and their generalized inverses are widely used inmathematics and other branches of science, such as control theory,system identification, programming theory, network theory,measurement, statistics and econometrics. Therefore, it is essential for usto understand the basic ideas and methods of blocked matrices andtheir generalized inverses. With the development of technologies,people are stepping into the information era. Blocked matrices andtheir generalized inverses have been widely used in practice, andbecome more and more important.The contents of this thesis are divided into three parts: In the first part,we discuss structural representation of generalized inverses of twomatrices. In the second part, we present the necessary and sufficientconditions for block independency of two matrices. In the last part, wediscuss the equivalent conditions of the necessary and sufficientcondition for block independency of two matrices.1.The structural representation for the generalized inverses of twomatricesIn Chapter two, we study the structural representation for thegeneralized inverses of two complex matrices A、C and A. By theC well-known quotient singular value decomposition(Q-SVD), we give thestructure of {1,2,3},{1,2,4} and {1,3,4}-inverses of A、C and A.C2.The necessary and sufficient condition for block ndependency oftwo matricesIn Chapter three, the the necessary and sufficient condition forblock independency of two matrices is studied. With respect to the definition the block independence given by Wang[38], we give thenecessary and sufficient condition for {1,2,3},{1,2,4} and {1,3,4}-inversesof two matrices.3.The equivalent conditions for block independency of twomatricesIn Chapter four, the equivalent conditions of the necessary andsufficient condition for block independency of the {1,2,3},{1,2,4} and{1,3,4}-inverses are further discussed under the two different definitions,and we proposed a new method for studying the block independenceof the generalized inverses of two matrices.