Products of exponentials of hermitian and complex symmetric matrices

A conjecture involving exponentials of Hermitian matrices is stated, and proved when one of the matrices has rank at most one. As a consequence, the complete conjecture is proved when the matrices are 2×2. A second conjecture involving exponentials of complex symmetric matrices is also stated, and completely proved when the matrices are 2×2. The two conjectures have similar structures but require quite different techniques to analyze.