Nonlinear Optimization Involving Polynomial Matrices and Their Generalized Inverses

摘要

This thesis examines the applications of the generalized inverse of a matrix. In particular, use is made of the generalized inverse of a matrix containing variable elements. Such matrices are referred to as multiparameter, polynomial, or variable element matrices. The notion of a generalized inverse in fact generalizes the concept of a matrix inverse. A matrix inverse exists only for square, non-singular matrices. The generalized inverse extends this notion to non-square, singular matrices. The classical matrix inverse, when it exists, is a unique element of the set of generalized inverses for the matrix. Many modern problems involve multiparameter matrices. The ability to obtain inverses for such matrices, both singular and non-singular, is a necessity in solving these problems. This thesis consolidates the theory of generalized inverses, including extensions to multiparameter matrices. An in depth discussion is made of the ST method for computing all generalized inverses of a matrix as well as the strong interface between the ST method and the Fundamental Theorem of Linear Algebra. Finally selected application problems are solved demonstrating the utility of the generalized inverse in such problems. (KR)