Weighted binary relations involving the Drazin inverse

Abstract

The Drazin inverse of a matrix has been used in the literature to define a preorder on the set of square complex matrices. In this paper we analyze new binary relations defined on the set of rectangular complex matrices and some relationships to the W -support idempotent. We introduce the class of weighted Drazin equal projectors and analyze the pre-orders on this class. Moreover, adjacent matrices are studied under the considered relations. Finally, some observations on weighted partial orders are given.

Relationships between DMP relation and some partial orders

“…Proof. If T has index at most 1 and T ≤ B, by the proof of Theorem 3.4, we have T 22 = 0 in (2)- (12) and (ii) By (9), (12) (13) and (14), (10), …”

“…So we omit the trivial cases T = 0 and B = 0. Some results related to binary relations defined by the group and Drazin generalized inverses can be found in [13].…”

Weighted binary relations for operators on Banach spaces

“…Some weighted pre-orders were defined on the set of rectangular matrices in [5]. These definitions were generalized to the class of bounded linear operators between Banach spaces in [8].…”

“…In [4], using the weighted Drazin inverse, new pre-orders and partial orders were introduced and investigated on the set of rectangular matrices, generalizing some results from [5].…”

Órdenes parciales y pre-órdenes definidos a partir de matrices inversas generalizadas.

“…La tesi finalitza amb un annexe on s'indiquen les conclusions finals i les línies futures d'investigació. Els resultats obtinguts en aquesta memòria es poden trobar en [31,32,33].…”