Norm conditions for convergence of infinite products (original) (raw)

On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

Discrete Dynamics in Nature and Society, 2018

We consider the problem of convergence to zero of matrix products AnBn • • • A1B1 with factors from two sets of matrices, Ai ∈ A and Bi ∈ B, due to a suitable choice of matrices {Bi}. It is assumed that for any sequence of matrices {Ai} there is a sequence of matrices {Bi} such that the corresponding matrix product AnBn • • • A1B1 converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, AnBn • • • A1B1 ≤ Cλ n , where the constants C > 0 and λ ∈ (0, 1) do not depend on the sequence {Ai} and the corresponding sequence {Bi}. Other problems of this kind are discussed and open questions are formulated.

Corrigendum/addendum to: Sets of matrices all infinite products of which converge

Linear Algebra and its Applications, 2001

This corrigendum/addendum supplies corrected statements and proofs of some results in our paper appearing in Linear Algebra Appl. 161 (1992) 227-263. These results concern special kinds of bounded semigroups of matrices. It also reports on progress on the topics of this paper made in the last eight years.

On the boundedness of infinite matrix products with alternating factors from two sets of matrices

arXiv (Cornell University), 2020

We consider the question of boundedness of matrix products AnBn • • • A1B1 with factors from two sets of matrices, Ai ∈ A and Bi ∈ B, due to an appropriate choice of matrices {Bi}. It is assumed that for every sequence of matrices {Ai} there is a sequence of matrices {Bi} for which the sequence of matrix products {AnBn • • • A1B1} ∞ n=1 is norm bounded. Some situations are described where in this case the norms of the matrix products AnBn • • • A1B1 are uniformly bounded, that is, AnBn • • • A1B1 ≤ C for all natural numbers n, where C > 0 is a constant independent of the sequence {Ai} and the corresponding sequence {Bi}. For the general case, the question of the validity of the corresponding statement remains open.

Sets of matrices all infinite products of which converge

Linear Algebra and its Applications, 1992

An infinite product IIT= lMi of matrices converges (on the right) if limi __ M, . . . Mi exists. A set Z = (Ai: i > l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing self-similar objects like von Koch's snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT= lAd,, where d = (d,, . , d,, . .>, which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of column-stochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set X is an RCP set.

On the Infinite Products of Matrices

Advances in Pure Mathematics, 2012

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  of matrices chosen from a possibly infinite set of matrices 0 i i

Infinite matrices and almost convergence

J. Indian Math. Soc, 1976

In the present paper, we characterize r q (u, p) : f ∞ , r q (u, p) : f and r q (u, p) : f 0 ; where f ∞ , f and f 0 denotes, respectively, the spaces of almost bounded sequences, almost convergent sequences and almost sequences converging to zero, where the space r q (u, p) of non-absolute type have recently been introduced by Neyaz and Hamid (see, [13]).

Infinite matrices and absolute almost convergence

In 1973, Stieglitz [5] introduced a notion of FB-Convergence which provided a wide generalization of the classical idea of almost convergence due to Lorentz [1]. The concept of strong almost convergence was introduced by Maddox [3] who later on generalized this concept analogous to Stieglitz's extension of almost convergence [4]. In the present paper we define absolute FB-Convergence which naturally emerges from the concept of FB-Convergence.

Some matrix transformations and almost convergence

Kathmandu University Journal of Science, Engineering and Technology, 2013

The sequence space bv(u,p) has been defined and the classes (bv(u,p):l?), (bv(u,p):c),and (bv(u,p):c0) of infinite matrices have been characterized by Ba?ar, Altay and Mursaleen ( see, [2] ). The main purposes of the present paper is to characterize the classes (bv(u,p):ƒ?),(bv(u, p):ƒ), and (bv(u,p):ƒ0), where ƒ?, ƒ, and ƒ0 denotes the spaces of almost bounded sequences, almost convergent sequences and almost convergent null sequences, respectively, with real or complex terms. Kathmandu University Journal of Science, Engineering and Technology Vol. 8, No. II, December, 2012, 89-92 DOI: http://dx.doi.org/10.3126/kuset.v8i2.7330