New Method for Computing the Generalized Inverses of a Matrix and Its Application to the Lyapunov Matrix Equation

摘要

This thesis examines a new method for computing four generalized inverses of a matrix. This method, the ST method, is based on the careful selection of a sequence of matrix multiplications and partitionings which provide a new foundation for computing four generalized inverses. Central to this approach is the partitioning of the two submatrices, R and C, where the product of their submatrices will give the generalized inverses of interest. Thus using this new representation, the generalized inverses of a matrix can be computed in a simple and direct manner. In this work, four generalized inverses are derived and computed in a systematic manner from this representation. These results are strongly tied to the solution of the matrix equation Ax=b where the general solution is given in terms of this new representation and computational technique. The computational technique is presented with a example and also as an algorithm. Included with the algorithm is an analysis of its computer implementation. The use of one generalized inverse is used to find the solution of a Lyapunov matrix equation.