查看“︁高登-湯普森不等式”︁的源代码

在物理學和數學上，'''高登─湯普森不等式'''（Golden–Thompson inequality）是一個由{{harvtxt|Golden|1965}}和{{harvtxt|Thompson|1965}}二氏所證明的不等式，該不等式的定義如下：

若<math>A</math>和<math>B</math>是[[埃尔米特矩阵]]，則以下不等式成立：

:<math> \operatorname{tr}\, e^{A+B} \le \operatorname{tr} \left(e^A e^B\right)
</math>

其中<math> \operatorname{tr}\, X </math>指的是矩陣的[[跡]]，而<math> e^X </math>則是[[矩阵指数]]。此不等式在[[统计力学]]上相當重要，而此不等式一開始也是由此而生的。

貝多蘭‧康斯坦多（Bertram Kostant）在1973年利用[[康斯坦多凸性定理]]（Kostant convexity theorem）將此不等式推廣到所有的緊緻李群（compact Lie group）之上。

==參考資料==

*{{Citation | last1=Bhatia | first1=Rajendra | title=Matrix analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94846-1 | id={{MathSciNet | id = 1477662}} | year=1997 | volume=169}}
* J.E. Cohen, S. Friedland, T. Kato, F. Kelly, ''Eigenvalue inequalities for products of matrix exponentials'', Linear algebra and its applications, Vol. 45, pp.&nbsp;55–95, 1982. {{doi|10.1016/0024-3795(82)90211-7}}
*{{Citation | last1=Golden | first1=Sidney | title=Lower bounds for the Helmholtz function | url=http://link.aps.org/doi/10.1103/PhysRev.137.B1127 | doi= 10.1103/PhysRev.137.B1127 | id={{MathSciNet | id = 0189691}} | year=1965 | journal=Phys. Rev. | series = Series II | volume=137 | pages=B1127–B1128}}
*{{Citation | last1=Kostant | first1=Bertram | title=On convexity, the Weyl group and the Iwasawa decomposition | url=http://www.numdam.org/item?id=ASENS_1973_4_6_4_413_0 | id={{MR|0364552}} | year=1973 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=6 | pages=413–455 | accessdate=2013-12-14 | archive-date=2018-02-05 | archive-url=https://web.archive.org/web/20180205184347/http://www.numdam.org/item?id=ASENS_1973_4_6_4_413_0 | dead-url=no }}
* D. Petz, [https://web.archive.org/web/20120212183206/http://www.renyi.hu/%7Epetz/pdf/64.pdf A survey of trace inequalities], in Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa 1994).
*{{Citation | last1=Thompson | first1=Colin J. | title=Inequality with applications in statistical mechanics | url=http://jmp.aip.org/jmapaq/v6/i11/p1812_s1 | doi=10.1063/1.1704727 | id={{MathSciNet | id = 0189688}} | year=1965 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=6 | pages=1812–1813 }}{{dead link|date=十一月 2017 |bot=InternetArchiveBot }}

==外部連結==
*{{citation|last=Tao|first=T.|url=http://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/|year=2010|title=The Golden–Thompson inequality|accessdate=2013-12-14|archive-date=2013-12-18|archive-url=https://web.archive.org/web/20131218143630/http://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/|dead-url=no}}(裡面有附上定理的證明)

{{DEFAULTSORT:Golden-Thompson inequality}}
[[Category:線性代數]]
[[Category:矩陣論]]
[[Category:不等式]]