Weighted Drazin Inverse And Singular Linear Equations Of Condition Number, And Their Condition Number

It is well-known that the generalized inverses of a matrix have wide applications in many areas such as differential and integral equations, operator theory, statics, optimal theory, control theory, Markov chains and etc. It has become one of the important studying fields in the world since the middle of the last century. In 1980 Cline and Greville gave the definition of the W-weighted Drazin inverse which is the extension of Drazin inverse. From then on many people studied the W-weighted Drazin inverse in different fields. In this paper we discuss the condition numbers of the W-weighted Drazin inverse and singular linear systems WAWx = b, b ∈ R((WA)k2), x∈R((AW)k1).First, various normwise relative condition numbers that measure the sensitivity of W-weighted Drazin inverse and the solution of singular linear systems are characterized.Second, we discuss the minimum quality of condition numbers of Ad,W and the singular linear systems.Third, since condition numbers can not be computed exactly, we give the definition of the condition number of the condition number to know the sensitivity of the problem "compute the condition number" .At last, we discuss the representation for the W-weighted Drazin inverse (A B)d,w of A B. Furthermore, the relation between the Kronecker product of the projector is established.