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Infinite product expansions for matrix n-th roots
Published online by Cambridge University Press: 09 April 2009
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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 then
converges and P(a) = S(a). Furthermore, if p(a) < r < 1, then
for all ν, where M depends on r and a but is independent of ν and q.
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- Copyright © Australian Mathematical Society 1968
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