Editing Matrix exponential (section)

== Inequalities for exponentials of Hermitian matrices ==
{{Main|Golden–Thompson inequality}}
For [[Hermitian matrix|Hermitian matrices]] there is a notable theorem related to the [[Matrix trace|trace]] of matrix exponentials.

If {{mvar|A}} and {{mvar|B}} are Hermitian matrices, then<ref>{{cite book | author=Bhatia, R. | title=Matrix Analysis |series=Graduate Texts in Mathematics|isbn=978-0-387-94846-1 | year = 1997 | publisher=Springer | volume=169}}</ref>
<math display="block">\operatorname{tr}\exp(A + B) \leq \operatorname{tr}\left[\exp(A)\exp(B)\right].</math>

There is no requirement of commutativity.  There are counterexamples to show that the Golden–Thompson inequality cannot be extended to three matrices – and, in any event, {{math|tr(exp(''A'')exp(''B'')exp(''C''))}} is not guaranteed to be real for Hermitian {{math|''A''}}, {{math|''B''}}, {{math|''C''}}.  However, [[Elliott H. Lieb|Lieb]] proved<ref>{{cite journal| doi=10.1016/0001-8708(73)90011-X | doi-access=free | last1=Lieb | first1=Elliott H. | authorlink1=Elliott H. Lieb | title=Convex trace functions and the Wigner–Yanase–Dyson conjecture | journal=[[Advances in Mathematics]]| volume=11 | pages=267–288 | year=1973|issue=3| url = http://www.numdam.org/item/RCP25_1973__19__A4_0/ }}</ref><ref>{{cite journal|doi=10.1007/BF01646492 | author=H. Epstein | title=Remarks on two theorems of E. Lieb | journal= Communications in Mathematical Physics|volume=31|pages=317–325 | year=1973| issue=4 | bibcode=1973CMaPh..31..317E | s2cid=120096681 | url=http://projecteuclid.org/euclid.cmp/1103859039 }}</ref> that it can be generalized to three matrices if we modify the expression as follows
<math display="block">\operatorname{tr}\exp(A + B + C) \leq \int_0^\infty \mathrm{d}t\, \operatorname{tr}\left[e^A\left(e^{-B} + t\right)^{-1}e^C \left(e^{-B} + t\right)^{-1}\right].</math>