Generalizations of some conditions for Drazin inverses of the sum of two matrices

Abstract

In this article, we present some formulas of the Drazin inverses of the sum of two matrices under the conditions P 2 QP = 0, P 2 Q 2 = 0, QPQ = 0 and PQP 2 = 0, PQ 2 = 0, QP 3 = 0 respectively. These conditions are weaker than those used in some literature on this subject. Furthermore, we apply our results to give the representations for the Drazin inverses of block matrix A B C D (A and D are square matrices) with generalized Schur complement is zero.

New Representation of Drazin Inverse for Block complex matrices

“…In 2011, the additive formula was given under the new condition P Q 2 = 0, P QP = 0 (see [9,Theorem 2.1]). This is an elementary result and it deduce many wider conditions under which the Drazin inverse is expressed, e.g., [7,10]. We denote by A π the eigenprojection of A corresponding to the eigenvalue 0 that is given by A π = I − AA D .…”

“…It is of interesting to find the Drazin inverse of the block complex matrix M. This problem is quite complicated and was expensively studied by many authors, see for example [3,4,9,10]. In Section 3, we apply these computational formulas to give the Drazin inverse of a block complex matrix M. If BCB = 0, DCB = 0, BCA(BC) π = 0 and DCA(BC) π = 0, we establish the representation of M D , which also extend the result of Yang and Liu (see [9,Theorem 3.1]).…”

Additive Properties of the Drazin Inverse for Matrices and Block Representations: A Survey

“…therefore we can see the connection between the Drazin inverse of the sum of two matrices and the Drazin inverse of 2 × 2 block matrix (see [27,Theorem 3.1]). Also, as it is presented in (2.1), the Drazin inverse of the sum P + Q can be expressed using the Drazin inverse of 2 × 2 block matrix P P Q I Q .…”

The matrix of matrices exponential and application

“…Computation of matrix exponential is required in many applications such as control theory [4,6], Markov chain process [25] and nuclear magnetic resonance spectroscopy [7]. For several years, great efforts have been devoted to the study of matrix [28,27,5] and matrix exponential (see [18,11,8]). A decently wide variety of methods for computing matrix exponential were introduced ( see for instance [16]).…”