# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 15A09 Matrix inversion, generalized inverses 15A24 Matrix equations and identities
Full Text:
##### References:
 [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