Partial Ordering Of Generalized Inverse And The Solution Of The Equation Of Matrix Polynomial Function

This thesis mainly studies the following problems:Partial ordering of matrices is one of the most discussed points on the matrix theory. Many specialists have been engaged in studying the partial ordering of matrices such as varied kinds of partial ordering and its applications to mathematical statistics. SVD is one of the most important and widely tools in Matrix Analysis. By the definition of partial ordering of matrices, some new definitions of partial ordering have been put forward, such as A ≤{1} B(?) AA{1} = BA{1}, A{1}A = A{1}B and A ≤{1,2} B(?) AA{1, 2} = BA{1, 2}, A{1, 2}A = A{1, 2}B. It is discussed four situations in detail, and sufficient and necessary conditions of the new partial ordering have been derived.Meanwhile in this dissertation, we redefine the solvability of the equation of matrix polynomial function, when the function f(X)is a general polynomial function. Using the definition and property of matrix function as well as Jordan canonical form, we discuss necessary and sufficient conditions of matrix polynomial functionf(X) = A over the field of R and C respectively, and give a procedure to solve it. We also give necessary and sufficient conditions that any solution can be explicitly expressed by matrix polynomial of A.