Representation of the function Tr(exp(A - λB)) as a Laplace transform with positive weight and some matrix inequalities
Representation of the function Tr(exp(A - λB)) as a Laplace transform with positive weight and some matrix inequalities
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将函数 Tr(exp(A - λB)) 表示为具有正权重和一些矩阵不等式的拉普拉斯变换
DOI：
10.1088/0305-4470/13/10/012
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
K. J. L. Couteur
中科院分区：
文献类型：
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作者：
K. J. L. Couteur
利用Bernstein定理，对有限厄米矩阵A和B考虑了Tr（exp(A- λ B)）可以写成一个正测度的拉普拉斯变换的猜想。给出了用exp(A)和B的元素的分差表示的矩的显式公式。对于一大类矩阵A和B，当A和B交换时，矩取最大值和最小值，从而建立了矩的上界和下界；进一步的分析表明，如果B是正定的，并且A和B是有界的，这通常是正确的。导出了指数的分差的几个不等式。此外，如果A和B都是正定的，则根据A和B的特征值推导出Tr（AnBn）和Tr(AB)n的上界和下界。
The conjecture that Tr(exp(A- lambda B)) can be written as a Laplace transform with positive measure rho is considered for finite Hermitian matrices A and B by means of Bernstein's theorem. An explicit formula is given for the moments of rho in terms of divided differences of exp(A) and elements of B. For a large class of matrices A and B the moments of rho take their maximum and minimum values when A and B commute and so upper and lower bounds for the moments of rho are established; further analysis suggests that this is generally true if B is positive definite and A and B are bounded. Some inequalities for the divided differences of the exponential are derived. Also, if A and B are both positive definite, upper and lower bounds are derived for Tr(AnBn) and Tr(AB)n in terms of the eigenvalues of A and B. Applications to problems of statistical mechanics and possibly Euclidean field theory are mentioned.