The exponential of a linear operator A (e.g. a matrix) is defined through its Taylor Seriespic.twitter.com/AE81YE1vQE
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As a result, for two genera linearl operators A & B, we have that eᴬeᴮ ≠ eᴬ⁺ᴮ. Rather, it is given by BCH formula below. The dots on the right indicate an infinite number of nested commutators ( e.g. [A, [A,B]] ). There is no known closed form for this expression.pic.twitter.com/kMFayF9vzZ
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So we see that eᴬeᴮ = eᴬ⁺ᴮ if and only if A and B commute.
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Baker-Campell-Hausdorff (BCH or CBH) formula: The theorem in Lie theory says that the logarithm of a noncommutative product exp(X) exp(Y) can be expressed in terms of iterated commutators of X and Y.
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