# Lie Product Formula

Let $A$ and $B$ be finite-dimensional Hermitian matrices, and let $v$ $:=$ $max${$||A||,||B||$}.

Show that $||(e^{-iAt/m}e^{-iBt/m})^m -e^{-i(A+B)t}||\leq\epsilon$ provided $m = \Omega(v^2t^2/\epsilon)$.

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Is that supposed to be a 2-norm? –  Guess who it is. Nov 26 '10 at 3:59
It is generally considered impolite to phrase your questions in the imperative voice. If this is homework, you should also indicate what you've done so far and where you've gotten stuck. It is not our job to do your homework for you every time you tell us to. –  Qiaochu Yuan Nov 26 '10 at 4:09

## 1 Answer

HINT: Consider the product of power series $\exp(A/m)\exp(B/m) = I + \frac{A}{m} + \frac{B}{m} + O\left(\frac{1}{m^2}\right)$. Then use this to estimate the logarithm for sufficiently large $m$, then exponentiate and raise to the $m$-th power.

This is from the proof in section 2.4 of "Lie Groups, Lie Algebras, and Representations" by Brian C. Hall.

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+1 for referencing this book! –  Matt Calhoun Nov 27 '10 at 1:44