What is the General Expression for the Product of Two Matrix Exponentials?

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SUMMARY

The discussion centers on the general expression for the product of two matrix exponentials, specifically for non-commuting matrices. The primary focus is on the relationship expressed through the Baker-Campbell-Hausdorff formula, which provides a way to express the product e^Ae^B in terms of the commutator [A,B]. This formula is essential for understanding the behavior of matrix exponentials when the matrices do not commute, as conventional methods only apply when [A,B] = 0.

PREREQUISITES
  • Understanding of matrix exponentials
  • Familiarity with commutators in linear algebra
  • Knowledge of the Baker-Campbell-Hausdorff formula
  • Basic concepts of Lie groups and Lie algebras
NEXT STEPS
  • Study the Baker-Campbell-Hausdorff formula in detail
  • Explore applications of matrix exponentials in quantum mechanics
  • Learn about Lie groups and their representations
  • Investigate numerical methods for computing matrix exponentials
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Mathematicians, physicists, and engineers working with linear algebra, particularly those involved in quantum mechanics or control theory, will benefit from this discussion.

dipole
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Is there an expression, in general, for the product of two matrix exponentials, for non-commuting matrices?

i.e. something of the form,
e^Ae^B = e^{( * )}

where the ( * ) would, I assume, depend in some way on the commutator [A,B]?

I can only find examples online when [A,B] = 0 which is rather trivial, but nothing about the more general case...
 
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