Some additive results on Drazin inverses☆
Section snippets
Background
Our aim is to investigate the existence of the Drazin inverse of the sum , where p and q are either ring elements or matrices. The Drazin inverse is the unique solution to the equationsfor some , if any. The minimal such k is called the index in(a) of a. If the Drazin inverse exists we shall call the element D-invertible.
An element a is called regular if for some x, and we denote the set of all such solutions by .
A ring with 1 is von Neumann (Dedekind)
D-inverses via powering
As a first example where powering can be used, we present the case where . We have Proposition 2.1 Suppose and ab are D-invertible and that . Then is D-invertible; and .
Proof
Using induction it is easily seen thatandIt is now straight forward to check that satisfies the necessary equations and . □
We note in passing that this result takes care of the example
Splittings
As always our starting point for the splitting approach is the factorization . Using Cline’s formula [1], we may writewhereandThere are two approaches that we can take, namely we can compute and then square the result, or we can directly compute or . We shall start by using the second approach.
Our first result is Theorem 3.1 Suppose that and are D-invertible, and
Converse results
We shall now assume that is D-invertible, and examine the D-invertibility of the related elements, and ba. We shall present one local result in addition to one global result. Proposition 4.1 Let . If has a Drazin inverse then so do and aba. Proof Using the notation of Proposition 3.1, we see that . Now if is D-invertible, then the matrices M and in (5), (6) are D-invertible, so that is D-invertible. Now is a LO splitting because .
Acknowledgement
The authors wish to thank an anonymous referee for his/her remarks and corrections.
References (10)
- et al.
Some additive results on Drazin inverses
Linear Algebra Appl.
(2001) - et al.
About the von Neumann regularity of triangular block matrices
Linear Algebra Appl.
(2001) - et al.
Drazin invertibility for matrices over an arbitrary ring
Linear Algebra Appl.
(2004) - R.E. Cline, An application of representation of a matrix, MRC Technical Report, 592,...
Pseudo-inverses in associative rings and semigroups
Am. Math. Monthly
(1958)
Cited by (0)
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Research supported by CMAT – Centro de Matemática da Universidade do Minho, Portugal, by the Portuguese Foundation for Science and Technology – FCT through the research program POCTI, and by Fundação Luso Americana para o Desenvolvimento (project # 273/2008).