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A Proof of Lie’s Product Formula

Gerd Herzog
The American Mathematical Monthly
Vol. 121, No. 3 (March 2014), pp. 254-257
DOI: 10.4169/amer.math.monthly.121.03.254
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Page Count: 4
Topics: Matrices
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A Proof of Lie’s Product Formula
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Abstract For d × d-matrices A, B and entire functions f, g with f(0) = g(0) = 1, we give an elementary proof of the formula \documentclass{article} \pagestyle{empty}\begin{document} $$\lim_{k \to \infty} (f(A/k) g(B/k))^k = \exp(f'(0)A+g'(0)B).$$ \end{document} For the case f = g = exp, this is Lie’s famous product formula for matrices.

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