矩陣論是線性代數的核心議題之一，對矩陣的分類及矩陣的性質的研究則是矩陣論的主要課題。近年來，矩陣在物理學及其他科學分支上的應用也相當地普遍。Golden-Thompson是一個在矩陣論上相當重要的不等式，它與矩陣的指數相關，且在統計力學上有其重要性；然而儘管Golden-Thompson對Hermitian矩陣而言是成立的，它一般而言是不成立的；此外，Hamiltonian矩陣也是一類在凝聚體物理學等物理學的一些分支上有其應用的矩陣。藉由討論實數Hamiltonian矩陣對跡的不等式，以及陶哲軒所提供的一篇關於Golden-Thompson不等式的證明，我在這篇文章嘗試指出Golden-Thompson不等式對Hamiltonian矩陣可能是不成立的；此外，由於矩陣在微分方程上亦有其應用，因此有可能如此的性質會影響某些微分方程的性質，本文亦提供了Floquet定理這個對交換矩陣而言成立的定理的證明及其應用。

The theory of Matrix is one of the central topics of linear algebra, the studies properties and classification of matrices is the main topic of the matrix theory. Recently, the use of matrices is common in physics and other science branches. Golden-Thompson inequality is an important inequality in Matrix theory, it is related to the exponential of matrices, and is important in statistical mechanics; however, although Golden-Thompson inequality holds for Hermitian matrices, it is not correct for all matrices in general; on the other hand, Hamiltonian matrices are also applied in some brances of physics like Condensed matter physics. By discussing inequalities of the traces of real Hamiltonian matrices, and the proof of the Golden-Thompson inequality provided by Terry Tao, I tried to show that Golden-Thompson inequality probably does not hold for Hamiltonian matrices in this thesis; besides, since matrices are also applied to differential equations, it is possible that similar properties may influence properties of some differential equations, the proof of the Floquet theorem for commutative matrices and its application to differential equations is also provided in this thesis.

1. Sze-Bi Hsu, Ordinary Differential Equations with Applications, World Scientific Publishing Co. Pte. Ltd. , Singapore, 2006
2. Terry Tao, The Golden-Thompson inequality, 15 July, 2010, taken from http://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/.
3. I.D. Coope, “On matrix trace inequalities and related topics for products of Hermitian matrix”, J. Math. Anal. Appl., 188, pp. 999–1001, 1994