定義矩陣上的三角函數恆等式之推廣

本論文的題目來源，是來自於科朗數學研究所所提供題目（Courant inst.）由作者Li Ta-Tsien編輯(1998)，而在Horn和Johnson(1985)所著作書中對於 e^A 用泰勒展開式表示，其中 A 為矩陣，即 I+A+A^2/2!+A^3/3!+⋯，此書也是利用泰勒展開式把三角函數推廣定義在矩陣上，我們主要探討在實數上的三角恆等式推廣到矩陣上是否成立，因此本論文重點是要研究它們的成立條件，不成立時，也用例子加以說明，若成立時，我們給予證明並用實際例子驗證它們。

關鍵字

矩陣；三角函數；推廣

並列摘要

The source of the topic of this thesis is from the Courant Institute of Mathematics (Courant inst.), edited by the author Li Ta-Tsien (1998). The book that was written by Horn and Johnson (1985), the scholars used e^A to express Taylor’s expansion, in which A is a matrix, and that is, I+A+A^2/2!+A^3/3!+⋯. This book also applied Taylor’s expansion to extend the definition of trigonometric functions to matrices. The aim was to explore whether the trigonometric identities on real numbers were extended to matrices. Therefore, the focus of this thesis was to study the conditions for their establishment. If they were not established, examples would be used to illustrate how trigonometric identities did not meet the conditions for establishment. If they were established, evidence and concrete example would also be provided to verify them.

並列關鍵字

Matrices ； Trigonometric Functions ； Extension

參考文獻

1. Horn, R. A. and Johnson, C. R.(1985), Matrix Analysis Cambridge University.

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2. Chen Ji-Xiu, Jiang Guo-Ying, Pan Yang-Lian, Qin Tie-Hu, Tong Yu-Sun, Wu Duan-Shui and Xu Sheng-Zhi(1998), Problems and Solutions in Mathematics, Edited by Li Ta-Tsien, World Scientific.