在本篇文章中，我們主要在討論隨著時間函數t變動且具有常數的秩 的Hamiltonain 矩陣，並探討其平滑的規格化正交分解。首先我們提供了 一個簡單的方法來證明 Hamiltonain 矩陣的SVD。再來，我們利用前面討 論的Hamiltonian矩陣的觀點來簡化 Takagi 分解的證明。最後，我們探討在最佳化控制系統中伴隨著 Hamiltonain 矩陣所發生的一些現象。 In this paper, we discuss some smooth orthonormal factorizations of smooth Hamiltonian matrix valued functions of constant rank. First, we will pro- vide a simple method to verify the singular value decomposition for Hamiltonian matrices of constant rank and then use the result to prove related decompositions. Second, we make use of the "Hamiltonian viewpoint" to give another proof of the Takagi's factorization. At last, we conclude a few facts in optimal control system where Hamiltonian matrices arise very often.
