On the Infinite Products of Matrices
Affiliation(s)
National Research Institute of Astronomy and Geophysics, Helwan, Egypt. Faculty of Science, Helwan University, Helwan, Egypt.
National Research Institute of Astronomy and Geophysics, Helwan, Egypt. Faculty of Science, Helwan University, Helwan, Egypt.
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.
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.
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
[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