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)
A note on the representations for the Drazin inverse of $2 \times 2$ block matrices. (English) Zbl 1121.15008
Given a matrix $M = \left[ {\matrix A & B \\ C & D \\ \endmatrix } \right]$, where $A \in {\Bbb C}^{m \times m} $ and $D \in {\Bbb C}^{n \times n}$, a new formula for the Drazin inverse $M^D$ is derived under the assumptions that: $A(I - AA^D)B = 0, C(I - AA^D)B = 0, BCAA^D = 0, DCAA^D = 0$.

MSC:
15A09Matrix inversion, generalized inverses
WorldCat.org
Full Text: DOI
References:
[1] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications. (2003) · Zbl 1026.15004
[2] Bru, R.; Climent, J.; Neumann, M.: On the index of block upper triangular matrices. SIAM J. Matrix anal. Appl. 16, 436-447 (1995) · Zbl 0827.15007
[3] Campbell, S. L.: The Drazin inverse and systems of second order linear differential equations. Linear and multilinear algebra 14, 195-198 (1983) · Zbl 0523.15007
[4] Campbell, S. L.; Meyer, C. D.: Generalized inverse of linear transformations. (1991) · Zbl 0732.15003
[5] Chen, X.; Hartwig, R. E.: The group inverse of a triangular matrix. Linear algebra appl. 237/238, 97-108 (1996) · Zbl 0851.15005
[6] Djordjević, D. S.; Stanimirović, P. S.: On the generalized Drazin inverse and generalized resolvent. Czechoslovak math. J. 51, No. 126, 617-634 (2001) · Zbl 1079.47501
[7] Gonzalez, N. Castro; Dopazo, E.; Robles, J.: Formulas for the Drazin inverse of special block matrices. Appl. math. Comput. 174, 252-270 (2006) · Zbl 1097.15005
[8] Hartwig, R. E.; Shoaf, J. M.: Group inverse and Drazin inverse of bidiagonal and triangular Toeplitz matrices. Austral. J. Math. 24A, 10-34 (1977) · Zbl 0372.15003
[9] Hartwig, R. E.; Li, X.; Wei, Y.: Representations for the Drazin inverse of a $2\times 2$ block matrix. SIAM J. Matrix anal. Appl. 27, 757-771 (2006) · Zbl 1100.15003
[10] Hartwig, R. E.; Wang, G.; Wei, Y.: Some additive results on Drazin inverse. Linear algebra appl. 322, 207-217 (2001) · Zbl 0967.15003
[11] Meyer, C. D.; Rose, N. J.: The index and the Drazin inverse of block triangular matrices. SIAM J. Appl. math. 33, 1-7 (1977) · Zbl 0355.15009
[12] Miao, J.: Results of the Drazin inverse of block matrices (in chinese). J. Shanghai normal univ. 18, 25-31 (1989)
[13] Wei, Y.: Expressions for the Drazin inverse of a $2\times 2$ block matrix. Linear and multilinear algebra 45, 131-146 (1998) · Zbl 0984.15004
[14] Wei, Y.; Li, X.; Bu, F.: A perturbation bound of the Drazin inverse of a matrix by separation of simple invariant subspaces. SIAM J. Matrix anal. Appl. 27, 72-81 (2005) · Zbl 1093.15008
[15] Wei, Y.; Li, X.; Bu, F.; Zhang, F.: Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices -- application of perturbation theory for simple invariant subspaces. Linear algebra appl. 419, 765-771 (2006) · Zbl 1151.15306