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Zbl 0998.15010
Wang, Guorong
The reverse order law for the Drazin inverses of multiple matrix products. (English)
[J] Linear Algebra Appl. 348, No.1-3, 265-272 (2002). ISSN 0024-3795

It is well known that the Drazin inverse has been widely applied to the theory of finite Markov chains and singular differential and difference equations. In a classic paper {\it T. N. E. Greville} [SIAM Rev. 8, 518-521 (1966; Zbl 0143.26303)] gave necessary and sufficient conditions of the reverse order law for the Moore-Penrose inverse $(AB)^{+}= B^{+}A^{+}$ to hold for two complex matrices $A$ and $B$. In general, the reverse order law does not hold for the Drazin inverse, that is $(AB)^{D} \ne B^{D}A^{D}$. Drazin proved that $(AB)^{D} = B^{D}A^{D}$ holds under the condition $AB = BA$. \par In the paper under review the author gives necessary and sufficient conditions for the $n$ term reverse order law $(A_1A_2\cdots A_n)^{D} = A_n^DA_{n-1}^D \cdots A_2^D A_1^D$ in terms of some rank equality.
[Nikolai I.Osetinski (Moskva)]

MSC 2000:: *15A09 Matrix inversion
15A24 Matrix equations

Keywords: Drazin inverse; Moore-Penrose inverse; reverse order law; matrix equation

Citations: Zbl 0143.26303

Cited in: Zbl 1079.15501

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