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 APM  Vol.2 No.5 , September 2012
On the Infinite Products of Matrices
Abstract: In different fields in space researches, Scientists are in need to deal with the product of matrices. In this paper, we develop conditions under which a product Пi=0∞ of matrices chosen from a possibly infinite set of matrices M={Pj, j∈J} converges. There exists a vector norm such that all matrices in M are no expansive with respect to this norm and also a subsequence {ik}k=0∞ of the sequence of nonnegative integers such that the corresponding sequence of operators {Pik}k=0∞ converges to an operator which is paracontracting with respect to this norm. The continuity of the limit of the product of matrices as a function of the sequences {ik}k=0∞ is deduced. The results are applied to the convergence of inner-outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.
Cite this paper: Y. Hanna and S. Ragheb, "On the Infinite Products of Matrices," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 349-353. doi: 10.4236/apm.2012.25050.
References

[1]   Y. S. Hanna, “On the Solutions of Tridiagonal Linear System,” Applied Mathematics and Computation, Vol. 189, 2007, pp. 2011-2016.

[2]   R. Bru, L. Elsner and M. Neumann, “Convergence of Infinite Products of Matrices and Inner-Outer Iteration Schemes,” Electronic Transactions on Numerical Analysis, Vol. 2, 1994, pp. 183-193.

[3]   N. K. Nichols, “On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations,” SIAM Journal on Numerical Analysis, Vol. 10, No. 3, 1973, pp. 460-469. doi:10.1137/0710040

[4]   P. J. Lanzkron, D. J. Rose and D. B. Szyld, “Convergence of Nested Classical Iterative Methods for Linear Systems,” Numerische Mathematik, Vol. 58, 1991, pp. 658- 702.

[5]   A. Frommer and D. B. Szyld, “H-Splittings and Two- Stage Iterative Methods,” Numerische Mathematik, Vol. 63, No. 1, 1992, pp. 345-356. doi:10.1007/BF01385865

[6]   I. Daubechifs and J. C. Lagarias, “Sets of Matrices All Infinite Products of Which Converge,” Linear Algebra and Its Applications, Vol. 161, 1992, pp. 227-263.doi:10.1016/0024-3795(92)90012-Y

[7]   R. Bru, L. Elsner and M. Neumann, “Models of Parallel Chaotic Iteration Methods,” Linear Algebra and Its Applications, Vol. 102, 1988, pp. 175-192.doi:10.1016/0024-3795(88)90227-3

[8]   L. Elsner, I. Koltracht and M. Neumann, “On the Convergence of Asynchronous Paracontractions with Application to Tomographic Reconstruction from Incomplete Data,” Linear Algebra and Its Applications, Vol. 130, 1990, pp. 65-82. doi:10.1016/0024-3795(90)90206-R

[9]   S. Nelson and M. Neumann, “Generalizations of the Projection Method with Applications to SOR Theory for Hermitian Positive Semidefinite Linear Systems,” Numerische Mathematik, Vol. 51, No. 2, 1987, pp. 123-141.doi:10.1007/BF01396746

[10]   R. S. Varga, “Matrix Iterative Analysis,” Prentice-Hall, Englewood Cliffs, 1961.

[11]   J. M. Optega and W. Rueinboldt, “Monotone Iterations for Nonlinear Equations with Application to Gauss-Seidel Methods,” SIAM Journal on Numerical Analysis, Vol. 4, No. 2, 1967, pp. 171-190. doi:10.1137/0704017

[12]   M. Neumann and R. J. Plemmons, “Convergent Nonnegative Matrices and Iterative Methods for Consistent Linear Systems,” Numerische Mathematik, Vol. 31, No. 3, 1978, pp. 265-279. doi:10.1007/BF01397879

[13]   M. Neumann and R. J. Plemmons, “Generalized Inverse-Positivity and Splittings of M-Matrices,” Linear Algebra and Its Applications, Vol. 23, 1979, pp. 21-35.doi:10.1016/0024-3795(79)90090-9

 
 
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