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Infinite product expansions for matrix n-th roots

Published online by Cambridge University Press:  09 April 2009

R. A. Smith
R. A. Smith
Affiliation:
Department of MathematicsUniversity of Durham, England
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In this paper a denotes a square matrix with real or complex elements (though the theorems and their proofs are valid in any Banach algebra). Its spectral radius p(a) is given by with any matrix norm (see [4], p. 183). If p(a) < 1 and n is a positive integer then the binomial series converges and its sum satisfies S(a)n = (1−a)−1. Let where q is any integer exceeding 1. Then u(a) is the sum of the first q terms of the series (2). Write and let a0, a1, a2,…be the sequence of matrices obtained by the iterative procedure Defining polynomials φ0(x), φ1(x), φ2(x),…inductively by we have aν = φν (a) and therefore aμaν = aνaμ for all 4 μ, ν. The following is proved in section 2: Theorem 1. If ρ(a) < 1 thenconverges and P(a) = S(a). Furthermore, if p(a) < r < 1, thenfor all ν, where M depends on r and a but is independent of ν and q.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 2 , May 1968 , pp. 242 - 249
Copyright
Copyright © Australian Mathematical Society 1968

References

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