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Abstract
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| The concept of Drazin inverse plays an important role in various fields like Markov chains, singular differential and difference equations, iterative methods, etc. A challenge of great interest in this area is to establish an explicit representation for the Drazin inverse of a 2x2 block matrix in terms of the Drazin inverses of the diagonal blocks, with arbitrary blocks. It was posed as an open problem by Campbell and Meyer in 1979, in conecction with the problem to find general expressions for the solutions of the second-order system of the differential equations. Starting from the general formula given by C. D. Meyer and N. J. Rose [6] for the Drazin inverse of triangular block matrices, an intensive research has been developed on this topic. Recently, some partial results have been obtained under specific conditions [1-5, 7]. In this paper, we provide an explicit formula for 2x2 block matrices assuming some conditions on the blocks. It generalizes results given by R. E. Hartwig, X. Li and Y. Wei [4] and by D. S. Djordjevic and P. S. Stanmirovic [3]. From our main result, some special cases are derived. References: [1] D. Cvetkovic-Ilic, A note on the representation for the Drazin inverse of 2 x 2 block matrices, Linear Algebra and its applications (2008), doi:10.1016/j.laa.2008.02.019. [2] N. Castro-Gonzalez, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 (2006) 252-270. [3] D. S. Djordjevic, P. S. Stanmirovic, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (2001) 617-634. [4] R. E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2x2 block matrix, SIAM J. Matrix Anal. Appl., 27 (2006) 757-771. [5] X. Li, Y. Wei, A note on the representations for the Drazin inverse of 2 x 2 block matrices, Linear Algebra Appl. 423 (2007) 332-338. [6] C. D. Meyer Jr., N. J. Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math 33 (1977) 1-7. [7] Y. Wei, Expression for the Drazin of 2 x 2 block matrix, Linear and Multilinear Algebra 45 (1998) 131-146. | |
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International
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Si |
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Congress
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15-th Conference of the International Linear Algebra Society ILAS 2008 |
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960 |
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Place
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Cancún |
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Reviewers
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Si |
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ISBN/ISSN
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Start Date
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16/06/2008 |
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End Date
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20/06/2008 |
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From page
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17 |
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To page
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18 |
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Abstracts 15-th Conference of the International Linear Algebra Society |