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The reverse order law for the Drazin inverses of multiple matrix products. (English) Zbl 0998.15010
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 $AB = BA$. 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.

15A09Matrix inversion, generalized inverses
15A24Matrix equations and identities
Full Text: DOI
[1] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications. (1974) · Zbl 0305.15001
[2] Campbell, S. L.; Meyer, C. D.: Generalized inverse of linear transformations. (1979) · Zbl 0417.15002
[3] Greville, T. N. E.: Note on the generalized inverse of a matrix product. SIAM rev. 8, 518-521 (1966) · Zbl 0143.26303
[4] Hartwig, R. E.: Block generalized inverses. Arch. rational mech., anal. 61, 197-251 (1976) · Zbl 0335.15004
[5] Marsaglia, G.; Styan, G. P.: Equalities and inequalities for rank of matrices. Linear multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003
[6] Tian, H.: On the reverse order laws (AB)D=BDAD. J. math. Res. exposition 19, No. 2, 355-358 (1999) · Zbl 0940.15003