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A note on the representations for the Drazin inverse of 2 × 2 block matrices

✍ Scribed by Xiezhang Li; Yimin Wei

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
126 KB
Volume
423
Category
Article
ISSN
0024-3795

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✦ Synopsis

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix M = A B C D in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of M, denoted by M D , has recently been obtained under the assumptions that C(I -AA D )B = O and A(I -AA D )B = O together with the condition that the generalized Schur complement D -CA D B be either nonsingular or zero. We derive an alternative representation for M D under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R(CAA D ) ⊂ N(B) ∩ N(D), where R(•) and N(•) are the range and null space of a matrix.

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