Research On The Generalized Bott-Duffin Inverse And Its Related Matrix Classes

We in this paper study characterizations and representations of the generalized Bott-Duffin inverse and its applications in solving restricted linear equations.Moreover,we discuss two kinds of matrix classes closely related to the generalized Bott-Duffin inverse,namely,the co-GBD matrices and the GBD matrices.To begin with,some new characterizations of the generalized Bott-Duffin inverse are given by its range and null space as well as matrix equations.Furthermore,the famous conclusion that rank equation characterizes the inverse of a nonsingular matrix is extended to the generalized Bott-Duffin inverse.On the basis of this result,several explicit expressions of the generalized Bott-Duffin inverse are shown.And,the limiting representations of the generalized Bott-Duffin inverse are also given.Then,we establish relations between the generalized Bott-Duffin inverse and two nonsingular bordered matrices,and obtain the Cramer’s rules for solutions of two restricted linear equations.Secondly,this paper focuses on discussing a matrix A satisfying that AA_（L）^{（（?））}-A_（L）^{（（?））}A is nonsingular,called a co-GBD matrix,where A_（L）^{（（?））}is the generalized Bott-Duffin inverse of A with respect to a subspace L.We mainly apply the PL-decomposition and subspace operations to give various characterizations of the co-GBD matrices.Based on the continuity of the generalized Bott-Duffin inverse,the continuity of the co-GBD matrices is considered.Additionally,we state several equivalent conditions for the nonsingularity of AA_（L）^{（（?））}+A_（L）^{（（?））}A and In-A(A_（L）^{（（?））})²A,which are closely related to the co-GBD matrices,and show inverses of their general versions in a decomposition form.Finally,we turn our attention to a GBD matrix A,that is,A satisfies AA_（L）^{（（?））}=A_（L）^{（（?））}A.This new matrix class is characterized from different perspectives in terms of matrix equalities,inclusion relations of subspaces,orthogonal projectors,and the P_L decomposition.Moreover,a explicit representation of an GBD matrix is given and its continuity is also considered.In addition,some numerical examples are given to illustrate the results obtained in this paper.