高登─汤普森不等式
在物理学和数学上,高登─汤普森不等式(Golden–Thompson inequality)是一个由Golden (1965)和Thompson (1965)二氏所证明的不等式,该不等式的定义如下:
若和是埃尔米特矩阵,则以下不等式成立:
其中指的是矩阵的迹,而则是矩阵指数。此不等式在统计力学上相当重要,而此不等式一开始也是由此而生的。
贝多兰‧康斯坦多(Bertram Kostant)在1973年利用康斯坦多凸性定理(Kostant convexity theorem)将此不等式推广到所有的紧致李群(compact Lie group)之上。
参考资料
- Bhatia, Rajendra, Matrix analysis, Graduate Texts in Mathematics 169, Berlin, New York: Springer-Verlag, 1997, ISBN 978-0-387-94846-1, MR1477662
- J.E. Cohen, S. Friedland, T. Kato, F. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear algebra and its applications, Vol. 45, pp. 55–95, 1982. doi:10.1016/0024-3795(82)90211-7
- Golden, Sidney, Lower bounds for the Helmholtz function, Phys. Rev., Series II, 1965, 137: B1127–B1128, doi:10.1103/PhysRev.137.B1127, MR0189691
- Kostant, Bertram, On convexity, the Weyl group and the Iwasawa decomposition, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 1973, 6: 413–455 [2013-12-14], ISSN 0012-9593, MR0364552, (原始内容存档于2018-02-05)
- D. Petz, A survey of trace inequalities, in Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa 1994).
- Thompson, Colin J., Inequality with applications in statistical mechanics, Journal of Mathematical Physics, 1965, 6: 1812–1813, ISSN 0022-2488, doi:10.1063/1.1704727, MR0189688[失效链接]
外部链接
- Tao, T., The Golden–Thompson inequality, 2010 [2013-12-14], (原始内容存档于2013-12-18)(里面有附上定理的证明)