The (2,2,0) Drazin inverse problem

Abstract

We consider the additive Drazin problem and we study the existence of the Drazin inverse of a two by two matrix with zero (2,2) entry.

A note on Cline’s formula for the generalized Drazin inverse

“…Cline's formula plays an important role in finding the formulae for the (generalized) Drazin inverse of a sum of two elements and the (generalized) Drazin inverse of a block matrix (see [5][6][7][8]). …”

Generalized inverses of a factorization in a ring with involution

“…The Moore-Penrose inverse of a (2, 2, 0) matrix Patrício and Hartwig [6,7] considered the Drazin inverse, group inverse of a (2, 2, 0) matrix over a ring. Inspired by this, we consider the MP-inverse of a (2, 2, 0) matrix.…”

General exact solutions of certain second-order homogeneous algebraic differential equations

“…In this case, the matrix Z has eigenvalues 0, α and β, where α and β are the different solutions of the equation where ϕ(t) = − β−1 α−β e αt + α−1 α−β e βt and ψ(t) = 1 α−β e αt − e βt . For any ξ = (v 11 , v 12 , v 13 ) T and η = (v 21 , v 22 , v 23 …”

“…It is worth noting that ever since 1980s, [2,3] it has been an open problem to find an explicit formula for the Drazin inverse of a general anti-triangular (operator) matrix. Although much progress has been made, [4][5][6][7][8][9][10][11][12] less has been done in the case that the Drazin index of the anti-triangular matrix turns out to be two, which is the concern of this paper.…”