The Reverse Order Laws For Generalized Inverses And The Eigenvalue Problem Of Updated Matrix

It is well known thatholds for the product of m nonsingular matrices A_i, i = 1,2,…, m. However, this so-called reverse order law is not necessarily true for generalized inverses. How to get the necessary and sufficient conditions of the reverse order law for generalized inverses of multiple matrix products is an important and interesting problem that is mndamental in the theory of generalized inverses of matrices and their applications.In this thesis, applying the maximal and minimal ranks of generalized Schur complement,we will discuss the equivalent conditions of the reverse order laws for generalizedinverses of multiple matrix products:where A_i{i,j, k} denotes the set of {i,j, k}-inverse of A_i and A_i∈C^{l_i×l_i+1}, i = 1,…, m. In Chapter 3, the necessary and sufficient conditions by some rank identities of known matrices for the following reverse order laws are derived:As the extensions of the above-mentioned results, Chapter 4 investigates the reverseorder laws for weighted generalized inverses of two matrix products. In Chapter 5 we discuss the forward order law and the mixed-type reverse order law for generalizedinverses of multiple matrix products. Since the necessary and sufficient conditions of the reverse order law,the forward order law and the mixed-type reverse order law for generalized inverses of multiple matrix products derived in Chapter 3,4,5 are only constituted by the rank equalities of the known matrices, the obtained results here are more simple and can be easily checked.In Chapter 6, by employing the minimal ranks of generalized Schur complements and some classic results, we discuss the inverse of a special Schur complement. As the applications of the reverse order laws, we present some explicit expressions for the inverse of such kind of Schur complement.Furthermore, in Chapter 7, we study the eigenvalues of a special rank-r updated complement matrix. By Leverrier's algorithm for m-D system, we provide an algorithmfor the eigenvalues of this matrix. Numerical example show that our algorithm is feasible and effective.