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In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials:

if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices $U$ and $V$, such that

$$ e^{iA}e^{iB} = e^{i (UAU^*+VBV^*)}.$$

My question is, how do I calculate/determine the $Us/Vs$ in the above representation?

For concreteness can someone show this using $2\times2$ or $3\times 3$ matrices? or any $N\times N$ matrices?

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    $\begingroup$ In general, the U and V here will not be unique (this can be seen by a simple dimension counting argument). The known proofs of the above statement are quite indirect and non-constructive; they prove that U and V exist, but do not give an explicit recipe for finding such a U and V. $\endgroup$
    – Terry Tao
    Commented Oct 4, 2013 at 3:26
  • $\begingroup$ @Terry: interesting (I had mistakenly presumed that this should be "essentially doable") --- this makes the question more interesting now! $\endgroup$
    – Suvrit
    Commented Oct 4, 2013 at 13:40
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    $\begingroup$ A partial computational result may be found in Section 4.2 of arxiv.org/pdf/0810.2656 --- perhaps that result can be extended to the general case. $\endgroup$
    – Suvrit
    Commented Oct 4, 2013 at 13:49

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