Geodesic
Convergence of infinite products of matrices and inner-outer iteration schemes
Electronic transactions on numerical analysis, Tome 2 (1994), pp. 183-193
We develop conditions under which a product T i=0 i of matrices chosen from a possibly infinite set of matrices S = Tj |j $\in J$ converges. We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence ik$\infty $of the sequence k=$0 \infty $of the nonnegative integers such that the corresponding sequence of operators Ti converges k k=0 to an operator which is paracontracting with respect to this norm. We deduce the continuity of the limit of the product of matrices as a function of the sequences ik$\infty $. But more importantly, k=0 we apply our results to the question of 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.
Classification : 65F10
Keywords: iterative methods, infinite products, contractions 
@article{ETNA_1994__2__a1,
     author = {Bru,  Rafael and Elsner,  L. and Neumann,  M.},
     title = {Convergence of infinite products of matrices and inner-outer iteration schemes},
     journal = {Electronic transactions on numerical analysis},
     pages = {183--193},
     year = {1994},
     volume = {2},
     zbl = {0852.65035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1994__2__a1/}
}
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Bru,  Rafael; Elsner,  L.; Neumann,  M. Convergence of infinite products of matrices and inner-outer iteration schemes. Electronic transactions on numerical analysis, Tome 2 (1994), pp. 183-193. http://geodesic.mathdoc.fr/item/ETNA_1994__2__a1/