First,an explicit representation A(2)T,S=(GA+E)^-1G of the outer invers A(2)T,S for a matrix A∈Cm×n with the prescribed range T and null space S is derived,which is simpler than A(2)T,S=(GA+E)^-1G-V(UV)-2UG prop...First,an explicit representation A(2)T,S=(GA+E)^-1G of the outer invers A(2)T,S for a matrix A∈Cm×n with the prescribed range T and null space S is derived,which is simpler than A(2)T,S=(GA+E)^-1G-V(UV)-2UG proposed by Ji in 2005.Next,a new algorithm for computing the outer inverse A(2)T,S based on the improved representation A(2)T,S=(GA+E)^-1G through elementary operations on an appropriate partitioned matrix GAInIn0 is proposed and investigated.Then,the computational complexity of the introduced algorithm is also analyzed in detail.Finally,two numerical examples are shown to illustrate that this method is correct.展开更多
The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linea...The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations(i.e.both least squares and the minimal norm)is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse solution are present.Motivated by recent research,important least square properties of composite outer inverses are collected.展开更多
Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surject...Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties： f＇（x0） is double splitting and R（f＇（x）） ∩ N（T0^＋） = {0} near x0. However, in infinite dimensional Banach spaces, f＇（x0） is not always double splitting （i.e., the generalized inverse of f（x0） does not always exist）, but its bounded outer inverse of f＇（x0） always exists. Only using the C^1 map f and the outer inverse To^# of f（x0）, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.展开更多
基金The National Natural Science Foundation of China(No.11771076).
文摘First,an explicit representation A(2)T,S=(GA+E)^-1G of the outer invers A(2)T,S for a matrix A∈Cm×n with the prescribed range T and null space S is derived,which is simpler than A(2)T,S=(GA+E)^-1G-V(UV)-2UG proposed by Ji in 2005.Next,a new algorithm for computing the outer inverse A(2)T,S based on the improved representation A(2)T,S=(GA+E)^-1G through elementary operations on an appropriate partitioned matrix GAInIn0 is proposed and investigated.Then,the computational complexity of the introduced algorithm is also analyzed in detail.Finally,two numerical examples are shown to illustrate that this method is correct.
基金P.S.Stanimirovic was supported by the Ministry of Education and Science,Republic of Serbia(Grant 174013/451-03-9/2021-14/200124)D.Mosic was supported by the Ministry of Education,Science and Technological Development,Republic of Serbia(Grant 174007/451-03-9/2021-14/200124)Y.Wei was supported by the bilateral project between China and Serbia The theory of tensors,operator matrices and applications(no.4-5)'.
文摘The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations(i.e.both least squares and the minimal norm)is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse solution are present.Motivated by recent research,important least square properties of composite outer inverses are collected.
基金Project supported by the National Natural Science Foundation of China (No. 10271053).
文摘Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties： f＇（x0） is double splitting and R（f＇（x）） ∩ N（T0^＋） = {0} near x0. However, in infinite dimensional Banach spaces, f＇（x0） is not always double splitting （i.e., the generalized inverse of f（x0） does not always exist）, but its bounded outer inverse of f＇（x0） always exists. Only using the C^1 map f and the outer inverse To^# of f（x0）, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.
